Linear Algebra Homework and Study GuidePhil R. Smith, Ph.D.August 22, 2011Homework Problem Sets Organized by Learning OutcomesTest I: Systems of Linear Equations; MatricesLesson 1An equation of n variables x1, x2, . . . , xnis said to be linear if it is in the forma1x1 + a2x2 ++ anxn=b wherea1, a2, . . . , an and b are real numberconstants.1. Give examples of linear equations using the denition.Generate at least three examples of linear equations on your ownand write them down in a list.2. Distinguish linear equations from nonlinear equations. A linear combination of unknowns(or variables) is the weighted sum ofthe unknowns, each term of which isthe product of one real number andone unknown. For example, 3x + 5y,2x1 3x2 +5x3, and ax + by + cz (witha, b, c real-valued constants) are linearcombinations of x, y, x1, x2, x3,and x, y, z respectively. A linearcombination of unknowns set equalto a constant is said to be a linearequation. Thus, the following are linearequations: 3x + 5y=11, 2x1 3x2 +5x3= 3, and ax + by + cz + d = 0(a) Which of the following are linear equations in x, y, and z?i. x23y +5z = 31ii. xyz = 12iii. 2x 3z = 2(b) Which of the following are linear equations in x1, x2, and x3?i. 2x113x2 +5x3= 1ii. x1= 5x2 +7x3iii. x1 + x12+12x3= 4iv. x1 +5x2 + x1x3= 8v. x3/51+2x24x3= 11vi. x1 +2x2 +23x3= 51/3(c) If k is a real-valued constant, which of the following are linearequations?i. x1x2 + x3= sin kii. kx11kx2= 10iii. 2kx1 +7x2x3= 03. Write solutions sets to linear equations using standard mathe-matical notation, including parametric notation for linear equa-tions with an innite number of solutions. The solution set of a linear equation isa list or description of the numbers thatmake the linear equation true. (a) Find the solution set of the following linear equations.i. 5x 7y = 4ii. 3x15x2 +4x3= 7iii. 3x18x2 +2x3x4 +4x5= 0(b) Find a linear equation in x and y that has the general solutionx = 5 +2t, y = t.linear algebra homework and study guide 2(c) Show that x=s, y=12s 52is also a general solution of theequation found in problem 3(b) above.4. Explain the interconnections between the terms linear combina-tion, linear equation, variable, unknown, and solution set.Create a diagram showing the relationships among the followingconcepts: linear combination, linear equation, variable, unknown, andsolution set.Figure 1: Example concept map. Majorconcepts are represented by squares,rectangles, circles, or ovals. Connectionsbetween concepts are represented withlabeled line segments or labeled arrows.Lesson 21. Give examples of systems of linear equations based on the de-nition.Generate several examples of linear systems and write them downin a list.2. Recognize and translate among notations for list, column, androw vectors.Complete the table of equivalent vectors below with the appropri-ate list, column, or row vectors.linear algebra homework and study guide 3(a) u = (5, 3, 2) = = .(b) v = =__2816__ = .(c) w = = =_7 13 19_.(d) r = (a, b, c, d, e) = = .(e) x = = =_x1x2x3x4x5x6x7_.3. Recognize and translate among formats for writing linear sys-tems. A system of linear equations:_2x +3y = 6x + y = 1A matrix equation Ax =b:_2 31 1_ _xy_ =_61_A column vector equationxc1 + yc2= b:x_21_+ y_31_ =_61_An augmented matrix _A[b:_2 3 61 1 1_In the problems below, a linear system has been described as eithera system of linear equations, a matrix equation, a column vectorequation, or an augmented matrix. Rewrite the each system belowin each of the remaining three formats.(a)__2 0 2 13 0 7 36 1 1 0__(b)__1 1 02 1 87 2 3____x1x2x3__ =__9103__(c) m__126__ + n__253__ + p__321__ =__642__(d)2y1+ 3y2+ 5y3+ 7y4= 5y1+ 5y3= 11y1+ 5y2+ 2y3 y4=1224y2+ 21y4= 154. State the denitions of inconsistent and consistent linear sys-tems.Write the denitions in your own words from memory. Be precise.5. Illustrate using two-dimensional graphs linear systems with nosolutions, one solution, and an innite number of solutions.Draw three 2-D graphs, one for each case.6. Identify the effects that coefcients have on solutions to lin-ear systems, specically how they can determine whether thesystem has no solutions, one solution, or an innite number ofsolutions.(a) For which value(s) of the constant k does the systemx y = 52x 2y = klinear algebra homework and study guide 4have no solutions? Exactly one solution? Innitely many solu-tions? Explain your reasoning.(b) Consider the system of equationsx + y + 2z = ax + z = b2x + y + 3z = cShow that for this system to be consistent, the constants a, b,and c must satisfy c = a + b.(c) For which value(s) of the constant a does the systemx + 2y 3z = 43x y + 5z = 24x + y + (a214)z = a +2have no solutions? Exactly one solution? Innitely many solu-tions? Explain your reasoning.Lesson 31. Write an arbitrary system of equations in general form. An arbitrary system of 3 lin-ear equations in 4 unknowns:___a11x1 + a12x2 + a13x3 + a14x4= b1a21x1 + a22x2 + a23x3 + a24x4= b2a31x1 + a32x2 + a33x3 + a34x4= b3An arbitrary system of m lin-ear equations in n unknowns:___a11x1 + a12x2 + + a1nxn= b1a21x1 + a22x2 + + a2nxn= b2............am1x1 + am2x2 + + amnxn= bmAn arbitrary augmented matrix ofm linear equations in n unknowns:__a11a12 a1nb1a21a22 a2nb2...............am1am2 amnbm__Write the system for r linear equations in s unknowns. Let theconstant c be the coefcient of the unknowns and let d be theconstant that follows the = sign.2. Write an arbitrary augmented matrix in general form.Write the augmented matrix for r linear equations in s unknowns.Let the constant c be the coefcient of the unknowns and let d bethe constant that follows the = sign.3. State the three elementary row operations.Write a description of the row operations in your own words frommemory.4. Review: Solve a 3-equation in 3-unknowns linear system usingintermediate algebra methods.Solve the following system using the methods you learned inintermediate algebra:___x y z = 13x y + z = 1x 2y z = 25. Gauss-Jordan by hand: Solve a 3-equation in 3-unknowns linearsystem using elementary row operations and the Gauss-Jordanelimination algorithm.Solve the following system using elementary row operations andthe Gauss-Jordan elimination algorithm:___x y z = 13x y + z = 1x 2y z = 2linear algebra homework and study guide 56. Technology - RREF command: Use technology to solve a 3-equation in 3-unknowns linear system.Solve the following system using a graphing calculator or othertechnology:___x y z = 13x y + z = 1x 2y z = 27. Determine whether a row of an augmented matrix has a leading1 or not. A zero row is a row of a matrix inwhich all the entries are zero.A nonzero row is matrix row thatcontains at least one nonzero entry.The rst nonzero element of a matrixrow is called a leading coefcient.If the leading coefcient is a 1, then it iscalled a leading 1 or pivot.Give three examples of row vectors with leading 1s and three rowswithout.8. State the three requirements for an augmented matrix to beconsidered in row-echelon form (REF).A matrix is said to be in row echelonformat (REF) if the rst three of thefollowing conditions are met and inreduced row echelon format (RREF) ifall four of the following conditions aremet:(a) If there are zero rows in the matrix,they are grouped at the bottom ofthe matrix.(b) If a matrix row is nonzero, then itsrst nonzero entry is a 1.(c) After the rst, the leading 1 of eachnonzero row occurs to the right ofthe leading entry of the previousrow.(d) In each column that contains aleading 1, all entries in the columnabove and below the leading 1 arezeros.In your own words and from memory, write the three require-ments for an REF matrix.9. State the four requirements for a matrix to be considered inreduced row-echelon form (RREF).In your own words and from memory, write the four requirementsfor an RREF matrix.10. Determine whether a matrix is in row-echelon form only (REFonly), reduced row-echelon form (RREF), or neither.(a)__1 0 00 1 00 0 1__(b)__1 0 00 1 01 0 0__(c)__1 2 00 1 00 0 0__(d)__0 1 00 0 10 0 0__(e)__1 5 20 0 10 0 0__(f)__1 3 40 0 00 0 1__(g)__1 4 30 1 10 0 0__(h)__0 0 00 0 00 0 0__(i)__1 0 3 10 1 2 4__ (j)__1 7 5 50 1 3 2__ (k)__1 0 0 50 0 1 30 1 0 4__linear algebra homework and study guide 6(l)__1 2 0 3 00 0 1 1 00 0 0 0 10 0 0 0 0__(m)__1 3 0 2 01 0 2 2 00 0 0 0 10 0 0 0 0__11. Given the RREF matrix for a system of linear equations, solve(read off) the solutions and write them in standard solutionnotation.The augmented matrices shown below have reduced by row oper-ations to RREF. Solve each system.(a)__1 0 0 30 1 0 00 0 1 7__(b)__1 0 0 7 80 1 0 3 20 0 1 1 5__(c)__1 3 0 00 0 1 00 0 0 1__(d)__1 6 0 0 3 20 0 1 0 4 70 0 0 1 5 80 0 0 0 0 0__(e)__1 0 0 90 1 0 10 0 1 30 0 0 0__12. Compare and contrast equivalent mathematical denitions.Consider the following alternative denition of REF: A matrix issaid to be in row echelon form if each of its nonzero rows has moreleading zeros than the previous row. Is this denition of row echelonform that same as the one you were given in class? If not, modifythis denition so that it is the same.Lesson 41. Apply the Gauss-Jordan elimination algorithm.(a) The linear system___x + y + z = 22x 2y z = 23x + y 2z = 2is being solved with theGauss-Jordan algorithm. For each step of the algorithm, statethe elementary row operation(s) that generated the step. Usecodes like R1 R3,12R3 R3, and 12R1 + R4 R4 torepresent elementary row operations.linear algebra homework and study guide 7i.__1 1 1 22 2 1 23 1 2 2__ii.__1 1 1 20 4 3 20 2 5 8__iii.__1 1 1 20 134120 2 5 8__iv.__1 1 1 20 134120 0 727__v.__1 1 1 20 134120 0 1 2__vi.__1 1 0 00 1 0 10 0 1 2__vii.__1 0 0 10 1 0 10 0 1 2__(b) The augmented matrix below is in the middle of the Gauss-Jordan elimination algorithm. Describe the next three row oper-ations. Theres no need to actually perform the row operations;just describe them using elementary row operation codes like2R1 + R3 R3.__1 2 1 2 4 3 110 3 5 1 2 4 60 0 1 4 4 4 50 0 0 4 3 1 4__(c) Solve each system with the Gauss-Jordan elimination algorithm.Document your transformations from one matrix to anotherwith elementary row operation codes like 34R2 R2.i.x1 + x2 +2x3= 8x12x2 +3x3= 13x17x2 +4x3= 10ii.2b +3c = 13a +6b 3c = 26a +6b +3c = 5linear algebra homework and study guide 8iii.2x13x2= 22x1 + x2= 13x1 +2x2= 1iv.5x12x2 +6x3= 02x1 + x2 +3x3= 12. Solve a linear system by learning the appropriate keystrokes ona graphing/symbolic calculator.To solve_1 1 11 1 3_ using a TI-89calculator, enter rref([1, 1, 1; 1, -1, 3]).Enter the rref function by selectingMatrix from the menu (by pressing 2ndMATH 4), followed by selection of rreffrom the menu (by pressing 4). Thenenter the matrix. Note that entries ineach row are separated by commas, andeach row is separated by a semicolon.Solve each system with a graphing/symbolic calculator. Writethe augmented matrix that was entered in the calculator, thentranscribe the resulting RREF matrix from your calculator screen,followed by a statement of the solution in usual mathematicalform.(a)x y +2z w = 12x + y 2z 2w = 2x +2y 4z + w = 13x 3w = 3(b)3x1 +2x2x3= 155x1 +3x2 +2x3= 03x1 + x2 +3x3= 116x14x2 +2x3= 30(c)x12x2 + x34x4= 1x1 +3x2 +7x3 +2x4= 2x112x211x316x4= 53. Generate and solve systems of linear equations from a variety ofmathematical contexts.(a) Solve the following system for x, y, and z.1x+ 2y 4z= 12x+ 3y+ 8z= 01x+ 9y 10z= 5(b) The following points lie on the cubic equation y= ax3+ bx2+cx + d: (0, 10), (1, 7), (3, 11), (4, 14). Find the coefcients a,b, c, and d and write the equation for the curve.(c) The following points lie on the circle ax2+ay2+bx +cy +d = 0:(2, 7), (4, 5), and (4, 3). Find the coefcients a, b, c, and dand write the equation for the circle.linear algebra homework and study guide 9Lesson 51. Dene a trivial solution to a system of linear equations. In mathematics, the word trivial isused to describe a solution or anexample of mathematical structurethat is ridiculously simple and/orimmediately obvious. For example,it takes no virtually no effort to seethat x =0, y =0 is a solution to3x +2y= 0 but more effort is requiredfor nontrivial solutions like x= 2, y =3 and x = 13, y =12.From memory and in your own words, state the denition of atrivial solution to a linear system of equations.2. Dene a nontrivial solution to a system of linear equations.From memory and in your own words, state the denition of anontrivial solution to a linear system of equations3. Give an example of a trivial solution.Write a trivial solution for a system with 8 unknowns.4. Give an example of a nontrivial solution.Write a nontrivial solution for a system with 7 unknowns.5. Give an example of a homogeneous system of linear equations.Write a specic homogeneous system of 3 equations in 4 un-knowns.6. Determine whether a homogeneous system has innitely manysolutions using the Number of Solutions to a HomogeneousSystem of Equations Theorem. Number of Solutions to a Homoge-neous System of Equations Theorem.A homogeneous system of linearequations with more unknowns thanequations has innitely many solutions.(a) Determine by inspection (i.e., no calculations with paper andpencil, or calculator) which of the following homogeneoussystems have nontrivial solutions.i.2x1 3x2+ 4x3 x4= 07x1+ x2 8x3+ 9x4= 02x1+ 8x2+ x3 x4= 0ii.x1+ 3x2 x3= 0x2 8x3= 04x3= 0iii.a11x1 + a12x2 + a13x3= 0a21x1 + a22x2 + a23x3= 0iv.3x12x2= 06x14x2= 0(b) Solve the following homogeneous systems of linear equationsby any method.i.2x1+ x2+ 3x3= 0x1+ 2x2= 0x2+ x3= 0linear algebra homework and study guide 10ii.3x1 + x2 + x3 + x4= 05x1x2 + x3x4= 0iii.2x y 3z = 0x +2y 3z = 0x + y +4z = 0iv.v +3w 2x = 02u + v 4w +3x = 02u +3v +2w x = 04u 3v +5w 4x = 07. Compare and contrast homogeneous and nonhomogeneous sys-tems of linear equations with more unknowns than equations.The Number of Solutions to a Homogeneous System of EquationsTheorem only applies only to homogeneous systems of equations.So, although it is true that every homogeneous linear system withmore unknowns than equations will have innitely many solu-tions, it is not necessarily the case for nonhomogeneous linear sys-tems. Find a nonhomogeneous linear system with more unknownsthan equations but does not have innitely many solutions.Lesson 61. Give the size of a matrix using the standard mathematical con-vention.Generate specic example matrices of size 2 3, 4 2, and 3 3.2. Give examples of square, column, and row matrices. (Note thatcolumn matrices are sometimes called column vectors, and rowmatrices are sometimes called row vectors.)Generate specic examples of a square matrix, a column matrix,and a row matrix.3. For a specic square matrix, identify the main diagonal and traceof the matrix.(a) if A =__6 1 32 4 57 9 37__, then compute tr(A).(b) if B =__7 34 89 2__, then compute tr(B).linear algebra homework and study guide 11(c) Solve the following equation tr(C) = 0 where C =__221 03 3 41 2 7__.4. Determine if two matrices are equal. Matrix Arithmetic Properties Theorem.If the sizes of the matrices A, B, and Care such that the stated operations canbe performed and k and l are any tworeal number constants, then(a) Commutative Property of Addition:A + B = B + A(b) Associative Property of AdditionA + (B + C) = (A + B) + C(c) Associative Property of Multiplica-tion A(BC) = (AB)C(d) Left Distributive Property:A(B + C) = AB + AC(e) Right Distributive Property:(A + B)C = AC + BC(f) (k + l)A = kA + l A(g) k(l A) = (kl)A(h) k(AB) = (kA)B = A(kB)(a) Use the denition of equal matrices to solve the following equa-tion for a, b, c, and d:_a b b + c3d + c 2a 4d_ =_8 17 6_(b) Are the matrices A and B equal? Why or why not?A =_1 25 7_, B =_1 2 05 7 0_5. Calculate scalar multiples of a matrix, add two matrices, andform linear combinations of matrices.Denition of Matrix Subtraction.For any two matrices A and B of thesame size, the difference of A and B isdened as: A B = A + (B)(a) Compute23A if A =__18 6 39 7 3612 6 5__.(b) Compute k2B if B =_0 11 3_.(c) Compute C + D if C =_3 52 1_ and D =_1 83 7_.(d) Compute E + F if E =_2 43 2_ and F =_1 3 13 4 1_.(e) Compute 2G3H if G =__1 0 21 1 12325__ and H=__3 1 11 4 03 023__.6. Generate the transpose of a matrix. Transpose Properties Theorem. If thesizes of the matrices are such that thestated operations can be performed,then(a) (AT)T= A(b) (A + B)T= AT+ BT(c) (kA)T= kAT, where k is any scalar(d) (AB)T= BTATGive the transpose of the following matrices A=_2 3 51 0 2_,B =__2178.5__, and C =__1 5 a1 b 93 3 c__.7. Determine when two matrices may be multiplied.(a) Let A be of size 3 4; B be size 3 4; C, 4 2; D, 3 2; and E,4 3. Determine which of the following products are dened.For those that are dened, give the size of the resulting matrix.i. AClinear algebra homework and study guide 12ii. BAiii. CDiv. ATB(b) Prove the following: If the matrix multiplications AB and BAare dened, then AB and BA are both square matrices.8. If it is possible to multiply two matrices, perform the calcula-tions by hand.(a)_0 11 2_ _3 21 0_(b)_x m2 1_ _4 pq 1_(c)__2 1 xy 1 32 z 1___23_(d)__8 1 a4 2 01 1 1____0bc__9. If it is possible to multiply two matrices, perform the calcula-tions by graphing/symbolic calculator.(a)__1 0 01 0 10 1 1____1 2 32 3 42 4 6__(b)__7 32 56 89 0___7 4 98 1 5_(c)_1 4 6 102 7 5 3___1 4 6 102 7 5 39 0 11 8__10. Determine the size of a matrix that is the result of some combi-nation of matrix operations.Let A be a p q matrix and let B be an s t matrix. Complete thefollowing statements in terms of the variables p, q, s, and t.(a) AB is dened provided that .(b) A + B2is dened provided that .(c) A3is dened provided that .(d) If BA is dened, then (BA)Tis a(n) matrix.(e) If B = BT, then .linear algebra homework and study guide 1311. Generate a matrix from a formula for an arbitrary entry aij.(a) Find the 4 4 matrix A=aij such that every arbitrary entrysatises the formula below:i. aij= i2+ jii. aij= ji1iii. aij=___1, if [i j[ > 11, if [i j[ 1(b) An identity matrix I is a matrix in which the diagonal entries(top,leftmost corner to bottom, rightmost corner) are all onesand all other matrix entries are zero. Write a formula for aij thatgenerates all n n identity matrices I.Lesson 71. Explain why matrix multiplication is not commutative. An identity matrix, designated I, is thematrix that when multiplied by anyother matrix returns the matrix. That is,AI= I A = A for all A.The inverse of matrix A, designatedA1, is the matrix that when multipliedby the original matrix returns theidentity. That is, AA1= A1A = I forall A.Explain mathematically why the matrix products AB and BA neednot be equal.2. Describe in symbolic form the following laws of matrix arith-metic: a) commutative law for addition, b) associative law foraddition, c) associative law for multiplication, d) left distributivelaw, and e) right distributive law.List the properties above from memory.3. Dene the zero matrix and describe its associated properties. A matrix A that has an inverse is saidto invertible. A matrix B without aninverse is said to be not invertible ornoninvertible. There is an alternateterminology that emphasizes thereverse. A matrix without an inverseis also said to be singular and a matrixwith an inverse would then be said tobe nonsingular. In general, I tend touse use invertible and not invertible todescribe matrices, but you should befamiliar with both sets of terminology.Give specic examples of zero matrices at the following sizes:2 2, 3 1, 1 4, and 2 3.4. Dene the identity matrix and give examples at various sizes.From memory, state the denition of the identity matrix. Write the3 3 identity matrix and the 5 5 identity matrix.5. Dene the inverse of a matrix.From memory, state the denition of the inverse of a matrix.6. Compute the inverse of a 2 2 matrix. The formula for computing the in-verse of 2 2 matrix is_a bc d_1=1ad bc_d bc a_.(a)_3 51 2_(b)_2 34 4_(c)_3 00 2_(d)_3 61 2_linear algebra homework and study guide 147. Describe the circumstances in which a matrix would not have aninverse. In other words, we multiply the cornersof the orginal matrix and subtract tond the determinant. Then the inverseis the reciprocal of the determinanttimes a new matrix in which a andd have exchanged places and we aretaking the opposite of the b and centries.Give all the possible types of 2 2 non-invertible matrices.8. Use properties of matrix operations, transpose, and inverse tosolve matrix equations.(a) A1=_2 31 5_(b) (5A)1=_1 23 7_(c) (7AT)1=_5 23 1_(d) (I +3A)1=_4 51 2_9. Evaluate functions in which matrices are taken as the inputs.If A =_2 13 1_, nd the value off (A) iff (x) = x2+3x 2.10. Describe how the inverse of a matrix product is related to theindividual inverses of each matrix. Inverse-Product Theorem. If A andB are invertible matrices of the samesize, the product of AB is invertibleand the inverse of the product is given by:(AB)1= B1A1. In other words, theinverse of the product is equal to thereverse product of the inverses.Inverse Matrix Properties Theorem. IfA is invertible and n is an integer suchthat n 0, then(a) A1is invertible and(A1)1= A(b) Anis invertible and(An)1= An= (A1)n.(c) kA is invertible for any scalar k ,= 0and(kA)1=1k A1(a) The Inverse-Product Theorem states that (AB)1=B1A1provided A and B are invertible and of the same size. Extendthis result to three matrices:Prove: If A, B, and C are invertible matrices of the same size,then (ABC)1= C1B1A1.(b) In the problem below, assume that all matrices are the same sizeand invertible. Solve for the X matrix.BCATXCBTA = ACTLesson 81. Explain the components of a deductive system of inquiry.Make a list of the components with a brief description of each.2. Compare and contrast a deductive system of inquiry like mathe-matics with an inductive system like chemistry or physics.Make a side-by-side chart comparing/contrasting the two systemsof inquiry.3. Describe the basic components of an If-Then proof.(a) Prove: The square of an even number is also even.linear algebra homework and study guide 15(b) Prove: If A and B are n n matrices, then tr (A + B)= tr (A) +tr (B).4. Identify symmetric matrices. A square matrix A is said to besymmetric if AT= A.(a) Which of the following matrices are symmetric?i._3 11 2_ii._7 33 0_iii.__5 1 81 2 08 0 11__iv.__0 1 87 0 45 3 0__v.__0 0 30 7 05 0 0__(b) Find all values a, b, and c for which A is symmetric.__2 a 2b +2c 2a + b + c3 5 a + c0 2 7__(c) Using as many variables a, b, c, ..., z as necessary, write formu-las for symmetric 2 2, 3 3, and 4 4 matrices.(d) Let A = [aij], B = [bij], C = [cij], and D = [dij] be n n matrices.Determine whether the matrices A, B, C, and D are symmetric.i. aij= i2+ j2ii. bij= ijiii. cij= i2 j2iv. dij= ij5. Establish properties of symmetric matrices through mathemati-cal proof.(a) Prove: If A is a square matrix, then A + ATis symmetric.(b) Prove: If B is a square matrix, then BBTis symmetric.(c) Prove: If A is a symmetric matrix, then A2is also a symmetricmatrix.i. Proof. Assume A is a symmetric matrix.ii. It follows that A = AT.linear algebra homework and study guide 16iii. In order to demonstrate that A2is symmetric, it sufces toshow that .iv. (A2)T= (AA)Tv. = ATATvi. = AAvii. = A2viii. Therefore, A2is symmetric by denition. Q.E.D.(d) Prove: If ATA = A, then A is a symmetric matrix.(e) Prove: If ATA = A, then A = A2.6. Establish properties of skew-symmetric matrices through mathe-matical proof. A square matrix A is said to beskew-symmetric if AT= A.(a) Prove: If A is an invertible, skew-symmetric matrix, then A1isalso a skew-symmetric matrix.(b) Prove: If A is skew-symmetric, then so is AT.(c) Prove: If A and B are skew-symmetric matrices, then so is A+B.(d) Prove: If C is a square matrix, then C CTis skew-symmetric.i. Proof. (write the 1st step of an if-then proof)ii. In order to demonstrate that C CTis skew-symmetric, itsufces to show that (C CT)T= (C CT).iii. (C CT)T= CT(CT)Tiv. = CTCv. = C + CTvi. = (C CT)vii. Therefore, C CTis skew-symmetric . Q.E.D.7. Describe how the inverse of the transpose of a matrix is relatedto the transpose of the inverse of a matrix. Inverse-Transpose Theorem. If A isan invertible matrix, then ATis alsoinvertible and (AT)1= (A1)T.Prove: If A is an invertible symmetric matrix, then A1is alsosymmetric.(a) Proof. Assume A is both invertible and symmetric.(b) It follows that A = AT.(c) In order to demonstrate that A1is symmetric, it sufces toshow that (A1)T= A1.(d) (A1)T= (AT)1(e) = A1(f) Therefore, A1is symmetric by denition of symmetric matri-ces. Q.E.D.linear algebra homework and study guide 17Lesson 91. Given an elementary matrix, state the associated elementary rowoperation. An elementary matrix, E, is simplyan identity matrix that has had oneelementary row operation appliedto it. Left-multiplying a matrix byan elementary matrix is the same asapplying the elementary row operationto it.Below is a specialized notation for thethree types of elementary matrices:(a) Eij represents an elementary trans-position matrix in which row iwas swapped with row j, e.g.,E23=__1 0 00 0 10 1 0__ will cause row 2and 3 to be swapped.(b) Ei(k) represents an elementarydiagonal matrix in which the lead-ing 1 in the ith row is multipliedby the constant k. This has theeffect of multiplying the ith rowby the given scalar. For example,E2(12)=__1 0 001200 0 1__ has the effectof multiplying the 2nd row by12.(c)iEj(k) represents an elementary rowreplacement matrix in which row i ismultiplied by scalar k and added torow j, e.g.,3E1(2) =__1 0 20 1 00 0 1__has the effect of multiplying row 3by 2 and adding the result to row1 to become the new row 1.For each elementary matrix below, give the associated row opera-tion (e.g., 2R1 + R3 R3) that would be performed if a givenmatrix was left-multiplied by the elementary matrix.(a) E14=__0 0 0 10 1 0 00 0 1 01 0 0 0__(b) E2 3_15_ =__1 0 00 1 00151__(c) E2(1) =_1 00 1_2. Explain how row operations can be represented as multiplicationsof elementary matrices.Consider the matrices A =__7 4 32 7 54 2 3__, B =__4 2 32 7 57 4 3__ andC =__7 4 32 7 53 2 0__.(a) What is the specic elementary matrix (use one of the followingnotations: Eij, Ei(k), or iEj(k)) that left-multiplies A to producematrix B?(b) What is the specic elementary matrix (use one of the followingnotations: Eij, Ei(k), or iEj(k)) that left-multiplies B to producematrix A?(c) What is the specic elementary matrix (use one of the followingnotations: Eij, Ei(k), or iEj(k)) that left-multiplies A to producematrix C?(d) What is the specic elementary matrix (use one of the followingnotations: Eij, Ei(k), or iEj(k)) that left-multiplies C to producematrix A?3. Recognize that an invertible matrix can be written as a productof elementary matrices.Write matrices A and B below as products of elementary matrices.(a) A =_3 43 1_linear algebra homework and study guide 18(b) B =__1 0 20 5 30 0 1__4. Find inverses of square matrices of higher order than 2 2.Using row operations to convert_A [ I_ _I [ A1_, ndthe inverse of the following matrix.A =__1 0 11 1 10 1 0__5. Find the inverse of square matrices of higher order than 2 2using a symbolic calculator.Find the inverses of the following matrices:(a) A =__3 1 01 2 25 0 1__(b) B =__2 4 8 96 5 6 10 2 1 57 9 0 1__(c) C =__1 5 21 1 70 3 4__Lesson 101. Solve a linear system by rst writing it in matrix form, thenmultiplying both sides by the inverse of the coefcient matrix. Unique Solution Theorem. If A is aninvertible square matrix (n n) and bis a column vector of constants (n 1),then the system of equations Ax=bhas exactly one solution, namely,x = A1b.(a) Solve the following systems of equations by inverting the coef-cient matrix and using the Unique Solution Theorem.i.4x13x2= 32x15x2= 9ii.x1 +3x2 + x3= 42x1 +2x2 + x3= 12x1 +3x2 + x3= 3iii.5x1 +3x2 +2x3= 43x1 +3x2 +2x3= 2x2 + x3= 5linear algebra homework and study guide 19(b) Solve the following general systems of equations by inverting thecoefcient matrix and using Unique Solution Theorem. Thenuse the resulting formula for x to nd the specic solution for aspecic b.i.3x1 +5x2= b1x1 +2x2= b2where b =_b1b2_ =_43_.ii.3x1 +5x2= b1x1 +2x2= b2where b =_b1b2_ =_21_.iii.x1 +2x2 +3x3= b12x1 +5x2 +5x3= b23x1 +5x2 +8x3= b3where b =__b1b2b3__ =__134__.iv.x1 +2x2 +3x3= b12x1 +5x2 +5x3= b23x1 +5x2 +8x3= b3where b =__b1b2b3__ =__500__.v.x1 +2x2 +3x3= b12x1 +5x2 +5x3= b23x1 +5x2 +8x3= b3where b =__b1b2b3__ =__113__.2. Solve a matrix equation by using the inverse of a matrix.Solve the following matrix equation for X:__1 1 12 3 00 2 1__ X =__2 1 5 7 84 0 3 0 13 5 7 2 1__3. Apply properties of diagonal matrices to solve various mathe-matical problems.(a) Determine whether the diagonal matrix below is invertible; if itis, nd the inverse by inspection.i._3 00 7_ii.__5 0 00 0 00 012__iii.__18 0 00 5 00 013__(b) The matrix products below include at least one diagonal matrixas a factor. Compute the product by inspection.i.__5 0 00 1 00 0 3____2 14 12 5__linear algebra homework and study guide 20ii._3 2 12 1 4___3 0 00 2 00 0 4__iii.__4 0 00 1 00 0 2____4 3 11 2 05 1 3____3 0 00 2 00 0 5__(c) Find A2, A2, and Akby inspection.i. Let A =_1 00 3_ii. Let A =__130 001400 012__linear algebra homework and study guide 21Test II: DeterminantsLesson 111. Count the number of inversions in a permutation and classifywhether the permutation is even or odd. A permutation is an arrangement of anite set of numbers. In a permutation,each element of the nite set can beused only one time. So _1 2 3_,_1 3 2_, and _3 1 2_ are exam-ples of permutations of the set 1, 2, 3but _1 1 2 3_ and _2 2 3_ arenot.For the permutations below, give the number of inversions andindicate whether the permutation is odd or even.(i)_3 1 2 4 5_ (ii)_1 2 3 4 5_ (iii)_5 1 2 4 3_(iv)_4 5 2 1 3_ (v)_2 1 5 4 3_ (vi)_3 1 5 4 2_(vii)_5 1 2 4 5_2. Find elementary products of a given matrix.Find all the elementary products of matrices B and C. Make surethat, in each elementary products, the factors are ordered with thei index in numerical order (e.g., b1b2b3or c1c2c3c4). An elementary product of a nnmatrix A is a product of n entries fromA, no two of which come from the samerow or the same column.(a) B =__b11b12b13b21b22b23b31b32b33__(b) C =__c11c12c13c14c21c22c23c24c31c32c33c34c41c42c43c44__3. Find signed elementary products of a given matrix. Each elementary product can be or-dered according to its rst index,e.g., a13a22a31. The j indices form anelementary products permutation,e.g., _3 2 1_. A signed elemen-tary product is simply the product of(1)# of inversions of the permutationand itsassociated elementary product, e.g.(1)3a13a22a31.The determinant of a square matrix isthe sum of all of its signed elementaryproducts.The determinant of a matrix A isdenoted either det (A) or [A[.For matrices B and C in the previous problem, nd all of thesigned elementary products.4. Calculate the determinant by using the denition.Use the denition to calculate each of the determinants below.Specically, in each case, nd all the elementary products, deter-mine their respective signs, and then add the signed products.(a)3 72 4(b)5 110 2(c)5 62 7(d)a 3 53 a 2(e)2 7 65 1 23 8 4linear algebra homework and study guide 22(f)3 0 02 1 51 9 4(g)c 4 32 1 c24 c 1 25. Calculate the determinant by using a graphing/symbolic calcula-tor.Use your TI-89 calculator or equivalent technology to check yourcalculations of the determinants of the matrices in the previousproblem.6. Solve equations involving determinants.(a) Find all values for which det (A) = 0 if A =_ +2 24 4_.(b) Find all values for which det (B) = 0 if B =__ 5 0 00 40 3 4__.(c) Solve for x:x 23 1 + x =2 x 61 0 31 3 x 37. Solve linear systems of equations using Cramers Rule. Cramers Rule If Ax= b is a system ofn linear equations in n unknowns suchthat det (A) ,=0, then the system hasthe unique solution:x1=det (A1)det (A)x2=det (A2)det (A)...xn=det (An)det (A)where Aj is the matrix formed whenthe jth column of A is replaced with theentries of b.(a)7x12x2= 33x1 + x2= 5(b)4x +5y = 211x + y +2z = 3x +5y +2z = 1(c)x 4y + z = 64x y +2z = 12x +2y 3z = 20(d)x13x2 + x3= 42x1x2= 24x13x3= 0(e)x14x2 +2x3 + x4= 322x1x2 +7x3 +9x4= 14x1 + x2 +3x3 + x4= 11x12x2 + x34x4= 4linear algebra homework and study guide 23Lesson 121. Calculate determinants by inspection. Basic Properties of DeterminantsZero Row/Column Property: If A is asquare matrix with a either a row orcolumn of zeros, then det (A) = 0.Proportional Rows Property: If A is asquare matrix with two proportionalrows or two proportional columns, thendet (A) = 0.Matrix Transpose Determinant The-orem. If A is a square matrix, thendet (AT) = det (A).Effects of Elementary Row Operationson the Determinant:(a) Multiplication of matrix row (orcolumn) by a scalar k multiplies thematrix determinant by that scalar k.(b) Switching two rows (or columns)of a matrix changes the sign of itsdeterminant.(c) Adding a scalar multiple of arow (or column) to another row(or column) has no effect on thedeterminant.(a) Use basic properties of determinants to calculate the determi-nants below. Dont use a calculator; look instead for patternsrelated to the basic properties of determinants.i.3 15 70 4 20 0 3ii.7 10 50 0 05 2 3iii.2 0 0 07 1 0 07 02 04 2 6 1iv.4 4 08 7 02 19 0v.2 1 31 7 42 1 3vi.1 2 32 4 65 7 5vii.127 31311 25613 5viii.1 0 0 00 1 0 00 0 7 00 0 0 1ix.0 0 0 10 1 0 00 0 1 01 0 0 0x.1 0 0 00 1 0 90 0 1 00 0 0 1linear algebra homework and study guide 24xi.0 0 0 10 0 1 00 1 0 01 0 0 0(b) Given thata b cd e fg h i= 5, nd the following determinants: Denition of Triangular Matrices.A matrix is said to be triangular if allthe entries below (or above) the maindiagonal are zero. A matrix in whichall of the nonzero entries are at orabove the main diagonal is said to bean upper triangular matrix; a matrixin which all of the nonzero entries areat or below the main diagonal is saidto be lower triangular. A matrix whichis simultaneously upper and lowertriangular is said to be diagonal.Triangular Matrix Determinant Theo-rem. If A is an upper triangular, lowertriangular, or diagonal matrix, then thedet (A) is simply the product of thediagonal entries of A.i.d e fg h ia b cii.3a 3b 3cd e f4g 4h 4iiii.a + g b + h c + id e fg h iiv.5a 5b 5cd e fg 2d h 2d i 2d2. Calculate determinants by reducing the matrix to row-echelonform.(a) Evaluate the matrix determinants below by reducing the matrixto row-echelon form.i.3 6 90 0 22 1 5ii.0 3 11 1 23 2 4iii.1 3 02 4 15 2 2iv.1 4 4 10 1 2 23 3 1 40 1 3 2v.1 0 0 30 1 2 22 3 2 30 3 3 3linear algebra homework and study guide 25(b) Use row reduction to establish the identity:1 1 1a b ca2b2c2= (b a)(c a)(c b)Lesson 131. Evaluate determinants of scalar multiplications, sums, differ-ences, products, and quotients where possible using propertiesof the determinant function. Determinant of a Product The-orem. If A and B are squarematrices of the same size, thendet (AB) = det (A) det (B).(a) Suppose that A and B are 3 3 matrices with det (A)= 5 anddet (B)= 2. Evaluate, if possible, the following determinantsusing the given specics and your knowledge of properties ofthe determinant function.i. det (A3)ii. det (B + A)iii. det (2B)iv. det (BTA)v. det (B1)(b) Suppose that C and D are 4 4 matrices with det (C)=13anddet (D)=7. Evaluate, if possible, the following determinantsusing the given specics and your knowledge of properties ofthe determinant function.i. det (CTC1)ii. det (2C +3D)iii. det (12D2)iv. det (DTCD)v. det (3C)12. Determine when a matrix is invertible by taking its determinant.Determinant Test for Invertibility The-orem A square matrix A is invertible ifand only if det (A) ,= 0.Use the Determinant Test for Invertibility Theorem to determinewhich of the following matrices is invertible:(a)__1 0 19 1 48 9 1__(b)__4 2 82 1 43 1 6__(c)__2 7 032 37 05 9 0__linear algebra homework and study guide 26(d)__3 0 15 0 68 0 3__3. Identify solutions to determinant equations by inspection. Matrix Inverse DeterminantTheorem. If A is invertible, thendet (A1) =1det (A).(a) Without actually computing the determinant, show that theequation below is satised by x = 0 and x = 2:x2x 22 1 10 0 5= 0(b) Without actually computing the determinant, show that:b + c c + a b + aa b c1 1 1= 0Lesson 141. Represent 2D and 3D vectors in drawings.(a) Sketch the vectors v1=(2, 5), v2=(3, 4), and v3=(0, 5)on the same set of xy-coordinate axes.(b) Sketch the vector w1=(3, 1, 5) within a set of xyz-coordinateaxes.(c) Sketch the vector w2=(2, 4, 6) within a set of x1x2x3-coordinate axes.2. Explain the effects of multiplying a vector by various scalarvalues.(a) What is the effect of multiplying the vector u = (1, 3, 4) by thefollowing scalars k? (i) k=2 (ii) k=1 (iii) k=12(iv) k=0(v) k = 12(vi) k = 1 (vii) k = 2(b) In general, what is the effect of multiplying any vector u byscalars in the following ranges?i. k > 1ii. k = 1iii. 0 < k < 1iv. k = 0v. 1 < k < 0vi. k = 1vii. k < 1linear algebra homework and study guide 273. Compute sums of vectors graphically and algebraically. Denition of Vector Addition. Twovectors u=(u1, u2) and v=(v1, v2)can be added algebraically by addingtheir respective components: u + v=(u1, u2) + (v1, v2)= (u1 + v1, u2 + v2).Triangle Law: If two vectors are rep-resented by two sides of triangle insequence (i.e, the tail of the second vec-tor begins at the arrowhead of the rst),then the third side of the triangle, inthe opposite direction of the sequence,represents the sum (or resultant) of thetwo vectors.Parallelogram Law: If two vectors arerepresented by two adjacent sides ofa parallelogram, then the diagonal ofthe parallelogram through the commonpoint represents the sum of the twovectors.Denition of Vector Inverse. A vectorx has an additive inverse in IRnif thereexists a vector y such that x +y= 0=y + x. The inverse vector y is denotedx. Furthermore if x=(x1, x2) is anyvector in IR2, then x = (x1, x2).Denition of Vector Subtraction.Subtracting a vector is the same asadding its inverse: x y = x + (y)Let u = (1, 4) and v = (2, 3).(a) Add the vectors u and v using the Triangle Law of Vector Addi-tion.(b) Add the vectors u and v using the Parallelogram Law of VectorAddition.(c) Add the vectors u and v algebraically by adding their compo-nents.4. Find linear combinations of vectors.(a) Calculate av + bw if a = 3, b = 2, v =_71_ and w =_22_.(b) Calculate av + bw if a = 1, b =4, v=_2 1 2_ andw =_3 1 5_.(c) Calculate av + bw if a=1.2, b=0, v=(1, 2, 5) and w=(4, 0, 152).(d) Find scalars k and l such that kv + lw=b where v=(2, 1),w = (1, 2), and b = (1, 0)(e) Let u=(2, 0, 4) and v=(1, 3, 6). Find values of k and l suchthat ku + lv = (5, 9, 14).(f) Find scalars c1, c2, and c3 such that c1u + c2v + c3w= b whereu = (3, 1, 2), v = (4, 0, 8), w = (6, 1, 4), and b = (2, 0, 4)5. Describe the geometry of a set of linear combinations of vectorsin IR2and in IR3. A locus is a set of points that containsall the points, and only the points, thatsatisfy the condition, or conditions,required to describe a geometric gure.For example, the locus of points in aplane that are all the same distance 3from a xed point would be a circle ofradius 3.Describe the locus of points contained in the following sets.(a) k(1, 1): k IR That is, all the scalar multiples of the vectoru = (1, 1) where the scalar ranges over the set of real numbers.(b) a(1, 0): a ZZ That is, all the scalar multiples of the vectoru = (1, 0) where the scalar ranges over the set of integers.(c) c(1, 0) : c 0, c IR(d) a(1, 0) + b(0, 1) : a ZZ and b IR(e) c(1, 0) + d(0, 1) : c, d IR(f) e(1, 1, 0) + f (0, 1, 1) : e, f IRlinear algebra homework and study guide 286. Calculate the Euclidean inner (dot) product and outer product oftwo vectors. The inner product of u and v:u v = uTv=_1 2 3__555__= (1)(5) + (2)(5) + (3)(5)= 30The outer product of u and v:u v = uvT=__123___5 5 5=__(1)(5) (1)(5) (1)(5)(2)(5) (2)(5) (2)(5)(3)(5) (3)(5) (3)(5)__=__5 5 510 10 1015 15 15__Recall that vectors can be written as column vectors. This allowsus to dene two types of vector multiplication, an inner productand an outer product. Consider two vectors u=(1, 2, 3) andv=(5, 5, 5). To nd the inner product, we rst view both vectorsas 3 1 column vectors u=__123__ and v=__555__. Its not possibleto multiply two 3 1 matrices. However, taking the transpose ofeither u or v would allow us to use regular matrix multiplicationto combine the two vectors. Note that, for uTv, we are multiplyinga 1 3 matrix times a 3 1 matrix to produce a 1 1 matrix, orscalar; and, for uvT, we are multiplying a 3 1 column matrix bya 1 3 row matrix to produce a 3 3 square matrix. The vectorproduct uTv is called the Euclidean inner product (or dot prod-uct) of u and v and is represented u v. The vector product uvTiscalled the outer product of u and v and is represented u v.For the vectors u and v below, calculate the dot product u v andthe outer product u v.(a) u = (2, 1) and v = (3, 7)(b) u = (3, 1, 6) and v = (2, 5, 3)(c) u = (u1, u2, u3) and v = (v1, v2, v3)7. Calculate the length of a vector. Denition of Magnitude of Vector:The magnitude, or length, of a vector vis given by [[v[[ = v v.Find the length of the following vectors:(a) u = (3, 1) (b) w = (1, 2, 5) (c) x = (4, 3) (d) v = (7, 4, 5)8. Derive relationships about vectors when their lengths are known.Magnitude of a Scalar-Vector ProductTheorem. [[kv[[ = [k[[[v[[.Denition of Unit Vector. A vector uis said to be a unit vector if [[u[[ =1.Sometimes, a hat-notation is usedto indicate a unit vector in the samedirection as a vector. For example, xwould represent a unit vector in thesame direction as x.(a) If [[u[[ =3 and [[v[[ =7, what are the largest and smallestpossible values for [[u v[[?(b) Let w = (2, 1, 5). Find all scalars k such that [[kw[[ = 4.(c) Prove: Unit Vector Theorem. If w is any nonzero vector, then w =1[[w[[w is a unit vector.(d) Use the result above to nd a unit vector that lies in the samedirection as w = (2, 5).(e) Use the result above to nd a unit vector that lies in the samedirection as w = (2, 6, 3).linear algebra homework and study guide 299. Determine whether any two vectors u and v are orthogonal, orwhether they have an included angle that is acute or obtuse. Angle Between Vectors Theorem. Ifu and v are vectors in either IR2or IR3,then u v = [[u[[[[v[[ cos ().Orthogonal Vectors Theorem. If u andv are vectors in either IR2or IR3, thenu v if and only if u v = 0.Acute/Obtuse Vectors Theorem. If uand v are vectors in either IR2or IR3and is the angle between them, then isacute if and only if u v>0 and isobtuse if and only if u v < 0.(a) Find the angle between vectors u and v.i. u = (2, 1) and v = (3, 7)ii. u = (3, 1, 6) and v = (2, 5, 3)(b) Without calculating the exact value of the angle, determinewhether the vectors u and v make an acute angle, make anobtuse angle, or are orthogonal.i. u = (6, 1, 4) and v = (2, 0, 3)ii. u = (0, 0, 1) and v = (1, 1, 1)iii. u = (6, 0, 4) and v = (3, 1, 6)iv. u = (2, 4, 8) and v = (5, 3, 7)(c) Let a = (2, k) and b = (3, 5). Find k such thati. a and b are parallel.ii. a and b are orthogonal.iii. the angle between a and b is3 .iv. the angle between a and b is4 .10. Calculate the projection of one vector onto another for 2D and3D vectors. Suppose that x and a are two vectors ineither IR2or IR3and a ,= 0. Then vectorx can be written as the resultant (sum)of two vectors x= x| +x where x| isa vector that is parallel to a and x is avector that is perpendicular to a.Note that because x| is parallel to thevector a, it can also be described as"lying on" a. For this reason, we alsocall x| the projection of x onto a andwrite x|= proja x.It can be shown that x = x| +x wherex|=proja x=x aa aa=x a[[a[[2a andx= x x|= x proja x = x x a[[a[[2aCalculate the following projections:(a) projau where u = (6, 3) and a = (3, 9).(b) projbw where w = (1, 2) and b = (2, 3).(c) projyx where x = (3, 1, 7) and y = (1, 0, 5).(d) projxy where y = (1, 0, 5) and x = (3, 1, 7).11. Given a nonzero reference vector a and an arbitrary vector x,rewrite the vector x as the sum of two vectors, one parallel to aand one orthogonal to a.Find x| and x such that x = x| +x for the given vectors x and abelow:(a) x = (2, 5) and a = (3, 1).(b) x = (2, 3, 1) and a = (1, 1, 1).(c) x = (3, 4, 5) and a = (1, 2, 3).12. Apply the concept of projection to nd the distance from a pointto a line. Applying the Magnitude of a Scalar-Vector Product Theorem to the projec-tion of x onto a, we get proja x=x a[[a[[2a = [x a[[[a[[2 [[a[[ = [x a[[[a[[.(a) Find the distance between the point (2, 1) and the line4x +2y +7 = 0.(b) Find the distance between the point (s, t) and the lineax + by + c = 0.linear algebra homework and study guide 3013. Establish properties of the dot product via mathematical proof.1 1The properties in this problem are truein both IR2and IR3. Once youve provedthe property in IR2(or IR3), it is easy towrite an analogous proof for the IR3(orIR2) case.(a) Prove: Symmetry Property. If u IR2and v IR2,then u v = v u.(b) Prove: Additivity Property. If u IR3, v IR3, and w IR3,then (u +v) w = u w+v w.(c) Prove: Homogeneity Property. If k is a real number scalar,u IR3, and v IR3, then (ku) v = k(u v).(d) Prove: Positivity Property. If v IR2, theni. v v 0ii. v v = 0 if and only if v = 0.Lesson 151. Calculate the cross product of two vectors. If u=(u1, u2, u3) and v=(v1, v2, v3)are vectors in IR3, then the cross prod-uct u v is dened asuv =_u2u3v2v3, u1u3v1v3,u1u2v1v2_Compute the vector cross products below by using the denition.(a) v w where v = (3, 2, 1) and w = (2, 5, 4).(b) wv where v = (3, 2, 1) and w = (2, 5, 4).(c) What is the relationship between v w and wv?2. Compare and contrast the dot product of two vectors with thecross product of two vectors.Make a table that lists similarities and differences between the dotproduct and the cross product.3. Given two vectors, nd a third vector that is orthogonal to both. The cross product of two vectors isorthogonal to each of its componentvectors: u v u and u v vFind a vector w that is orthogonal to the vectors u and v. Thenverify orthogonality by showing that u w = 0 and v w = 0.(a) u = (3, 1, 4) and v = (2, 1, 3).(b) u = (2, 0, 5) and v = (1, 1, 7).4. Find the area of a parallelogram spanned by two vectors. Area(parallelogram spanned by u & v)= [[u v[[(a) Find the area of the parallelogram spanned by u=(1, 1, 2)and v = (3, 1, 4).(b) Find the area of the parallelogram determined by the fourpoints P(1, 2), Q(2, 1), R(6, 1), and S(3, 4).5. Find the area of a triangle spanned by two vectors. Area(triangle spanned by u & v) =12[[u v[[(a) Find the area of the triangle spanned by u=(3, 1, 3) andv = (2, 2, 2).(b) Find the area of the triangle determined by the three pointsA(0, 4, 1), B(6, 2, 5), and C(10, 10, 10)linear algebra homework and study guide 316. Given a xed point and a vector, nd the equation of the linethat passes through the xed point and is orthogonal to thevector. Equation for a Line or a Plane Givena Fixed Point and a Normal Vector: In2-space, let n=(a, b); let P be a xedpoint (x0, y0); and X be an arbitrarypoint (x, y). In 3-space, let n=(a, b, c);let P be a xed point (x0, y0, z0); and letX be an arbitrary point (x, y, z). Thenthe equation of the line/plane is givenby:n PX = 0_ax + by + c = 0 (a line in IR2)ax + by + cz + d = 0 (a plane in IR3)Find the equation of a line ax + by + c= 0 that passes through Pand is orthogonal to n.(a) P(3, 1) and n = (5, 1)(b) P(2, 5) and n = (3, 4)7. Given a xed point and a vector, nd the equation of the planethat passes through the xed point and is orthogonal to thevector. Find the equation of a plane ax + by + cz + d =0 thatpasses through P and is orthogonal to n.(a) P(1, 3, 2) and n = (2, 1, 1)(b) P(3, 0, 0) and n = (0, 0, 1)(c) P(0, 0, 0) and n = (3, 2, 1)8. Find the equation of the plane that is determined by three givenpoints.(a) P(5, 3, 4), Q(2, 1, 7), and R(1, 4, 5)(b) P(2, 0, 1), Q(1, 2, 3), and R(3, 1, 2)Parametric Equations for a Line Givena Fixed Point and a Direction Vector:In 2-space, let v = (q, r) be the directionvector for the line; let P be a xedpoint (x0, y0) on the line; and X be anypoint (x, y) on the line. In 3-space, letv=(q, r, s) be the direction vector;let P be a xed point (x0, y0, z0) on theline; and let X be any point (x, y, z) onthe line. Then parametric equations ofthe line/plane can be derived from thevector equation:X = vt + P (x, y) = (q, r)t + (x0, y0) (x, y) = (qt + x0, rt + y0)_x = qt + x0y = rt + y0orX = vt + P (x, y, z) = (q, r, s)t + (x0, y0, z0) (x, y, z) = (qt + x0, rt + y0, st + z0)___x = qt + x0y = rt + y0z = st + z09. Given a xed point and a direction vector, nd parametric equa-tions for a line in 2- and 3-space.(a) Find the parametric equations x= qt + x0, y= rt + y0 of a linethat passes through P and is parallel to v = (q, r).i. P(2, 3) and v = (3, 1)ii. P(1, 2) and v = (5, 1)iii. Find parametric equations for the line passing through pointsP(3, 5) and Q(1, 8).(b) Find the parametric equations x= qt + x0, y= rt + y0, z= st +z0 of a line that passes through P and is parallel to v = (q, r, s).i. P(2, 3, 1) and v = (3, 1, 2)ii. P(2, 2, 5) and v = (1, 0, 1)iii. Find parametric equations for the line passing through pointsP(4, 2, 3) and Q(2, 7, 4).10. Analyze geometric relationships between lines and planes byrecognizing normal vectors implicit in equations of the formax +by +c = 0 and ax +by +cz +d = 0 and recognizing directionvectors for lines in sets of parametric equations like x= qt + x0,y = rt + y0 and x = qt + x0, y = rt + y0, z = st + z0(a) Determine whether the pairs of planes below are parallel.linear algebra homework and study guide 32i. 5x 3y +7z = 12 and 3x +9y 4z = 13ii. 6x = 24y 12z +18 and12x 2y + z =32(b) Determine whether the given line and plane are parallel.i. x = 5 4t, y = 1 t, z = 3 +2t and x +2y +3z 9 = 0ii. x = 3s, y = 2s +1, z = s +2 and 4x y +2z = 1(c) Determine whether each pair of planes is perpendicular.i. 3x y + z = 4 and x +2z = 1,ii. x 2y +3z = 4 and 2x +5y +4z = 1(d) Determine whether the given line and plane are perpendicular.i. x = 2 4t, y = 3 2t, z = 1 +2t and 2x + y z 5 = 0ii. x = q +2, y = q +1, z = 3q +5 and 6x +6y = 7(e) Find parametric equations for the line of intersection of theplanes 7x 2y +3z = 2 and 3x + y +2z +5 = 0.(f) Find an equation of the plane that passes through the point(3, 6, 7) and is parallel to the plane 5x 2y + z 5 = 0.(g) Find the point of intersection of the line x = 5t +9,y = t 1, z = t +3 and the plane 2x 3y +4z +7 = 0.Lesson 161. Explain any limitations to graphing vectors in spaces with di-mensions higher than 3.If possible, sketch a graph of the vector v =(3, 2, 1, 5, 1) ona set of x1x2x3x4x5 axes. If not, explain any limitations and/orworkarounds to graphing vectors in IR5.2. Compute linear combinations of vectors in IRn.(a) Find ku where k =13and u = (1, 0, 1, 9).(b) Find 2v 3w where v = (2, 1, 1, 3, 7) and w = (4, 2, 1, 3, 3).(c) Find scalars k1, k2, k3, and k4 such that k1(1, 3, 2, 0) +k2(2, 0, 4, 1) +k3(7, 1, 1, 4) + k4(6, 3, 1, 2) = (0, 5, 6, 3).3. Calculate the dot product in IRn.(a) Find u v if u = (1, 0, 1, 3, 2, 1) and v = (0, 2, 4, 1, 2, 3).(b) Find x y if x = (1, 7a, 2b, 3, 0) and y = (0, 2, 1, 1, 4c).4. Calculate and use the norm (length) of vectors in IRn.(a) Find [[u[[ if u = (1, 4, 1, 5, 2, 1).(b) Find all scalars k such that [[k(2, 3, 0, 6)[[ = 5.linear algebra homework and study guide 33(c) Prove: If v is a nonzero vector in IRn, thenv[[v[[has length 1.(d) Find the distance between u = (0, 2, 1, 1) and v = (3, 2, 4, 4).(e) Find the distance between x =(3, 3, 2, 1, 0) and y =(1, 5, 0, 4, 2).5. Determine whether two vectors in IRnare orthogonal.(a) u = (2, 4, 3, 0, 2, 0) and v = (0, 5, 4, 1, 1, 12).6. Establish properties of the dot product in IRnvia mathematicalproof.(a) Prove: If k IR and u IRn, then [[ku[[ = [k[[[u[[.(b) Prove: If u IRn, v IRn, and w IRn,then (u +v) w = u w+v w.linear algebra homework and study guide 34Test III: General Vector SpacesLesson 171. Dene a real vector space. A vector space is a nonempty set V ofobjects that are called vectors. On thisset, two operationsvector additionand scalar multiplicationare dened.Let x, y, z V and , IR.I. Closure Property of Addition:If x V and y V, then x +y VII. Commutative Property:x +y = y +xIII. Associative Property:x + (y +z) = (x +y) +zIV. Additive Identity Property:0 V such that x +0 = x, x VV. Additive Inverse Property:x V (x) V such thatx + (x) = 0VI. Closure Property of Scalar Multi-plication:If IR and x V, then x VVII. Vector Distributive Property:(x +y) = x +yVIII. Scalar Distributive Property:( + )x = x + xIX. Scalar Associative Property:(x) = ()xX. Scalar Identity Property:1x = xState from memory the 10 axioms of the denition of a real vectorspace.2. Determine whether a set of objects plus two operations is a realvector space.To show that something is a real vectorspace all 10 axioms must be satised.To show something is not a real vectorspace, nd at least one axiom that failsto hold.In the exercises below, you are given a set of objects, V, and twooperations called vector addition (represented with +) and scalarmultiplication (represented by juxtaposition). Determine whichsets are under the operations form a vector space by verifying thatall 10 properties of the vector space denition hold, or show that atleast one real vector space axiom fails to hold.(a) Let V= IR2be the set of objects. In other words, V is the set ofall order pairs (x, y) of real numbers. Vector addition and scalarmultiplication are dened as follows:u +v = (x, y) + (x/, y/) = (x + x/, y + y/)ku = k(x, y) = (x, ky)(b) Let V= (x, y) IR2: x 0 be the set of objects. Vectoraddition and scalar multiplication are dened in the standardway for two-dimensional vectors.(c) Let V=IR3be the set of objects. In other words, V is the setof all order triples (x, y, z) of real numbers. Vector addition andscalar multiplication are dened as follows:u +v = (x, y, z) + (x/, y/, z/) = (x + x/, y + y/, z + z/)ku = k(x, y, z) = (0, 0, 0)(d) Let V=IR2be the set of objects. Vector addition and scalarmultiplication are dened as follows:u +v = (x, y) + (x/, y/) = (x + x/, y + y/)ku = k(x, y) = (2kx, 2ky)(e) Let V=__x 11 y_ : x, y IR_ be the set of objects. That is, Vis the set of all 2 2 matrices such that the antidiagonal entriesare all 1. Vector addition is dened as the standard way ofadding two matrices, and scalar multiplication is dened as thestandard way for multiplying a matrix by a scalar.linear algebra homework and study guide 35(f) Let V=__x yz 0_ : x, y, z IR_ be the set of objects. That is, Vis the set of all 2 2 matrices such that bottom right corner is0. Vector addition is dened as the standard way of adding twomatrices, and scalar multiplication is dened as the standardway for multiplying a matrix by a scalar.(g) Let V=IR2be the set of objects. Vector addition and scalarmultiplication are dened as follows:u +v = (a, b) + (c, d) = (a + c +1, b + d +1)ku = k(a, b) = (ka, kb)(h) Let V= (1, t): t IR be the set of objects. Vector additionand scalar multiplication are dened as follows:u +v = (1, ) + (1, ) = (1, + )ku = k(1, ) = (1, k)(i) Let V= (x, y) IR2: 2x +3y= 1 be the set of objects. Vectoraddition and scalar multiplication are dened in the standardway for two-dimensional vectors.(j) Let V= (x, y) IR2: 2x +3y= 0 be the set of objects. Vectoraddition and scalar multiplication are dened in the standardway for two-dimensional vectors.(k) Given the two problems above, what can you say about V=(x, y) IR2: ax + by = c3. Establish additional properties of real vector spaces using math-ematical proof.(a) Prove: If V is a vector space, u V, and k IR, then k0 = 0.(Justify each step of the proof below with the appropriate math-ematical property, theorem, or denition.)i. Proof. Assume V is a vector space, u V, and k IR.ii. Note that k0 + ku = k(0 +u)iii. = kuiv. We know that ku V.v. And if ku V, then ku V.vi. Consider the sum (k0 + ku) + (ku).vii. (k0 + ku) + (ku) = ku + (ku)viii. k0 + (ku + (ku)) = ku + (ku)ix. k0 +0 = 0x. Therefore, k0 = 0xi. Q.E.D.linear algebra homework and study guide 36(b) Prove: If u, v, and w are vectors in a vector space V, andu +w = v +w, then u = v.i. Proof. Assume u, v, and w are vectors in a vector space V andu +w = v +w, then u = v.ii. Since w is a vector in V, then so is w.iii. It follows that the sums (u +w) + (w) and (v +w) + (w)are also in V.iv. (u +w) + (w) = (v +w) + (w)v. Simplifying the left hand side (LHS) of the equation above:(u +w) + (w) = u + (w+ (w))vi. = u +0vii. = uviii. Simplifying the right hand side (RHS) of the equation in (iv):(v +w) + (w) = v + (w+ (w))ix. = v +0x. = vxi. It follows that u = v.xii. Q.E.D.(c) Prove: There exists one and only one zero vector 0.i. Proof. Suppose to the contrary that there exists more than onezero vector.ii. It would follow that there were at least two different zerovectors, say 01 and 02, and that 01 ,= 02.iii. Consider the sum 01 +02iv. Since 02 is a zero vector in V,then 01 +02= 01v. But 01 is also a zero vector in V,so 01 +02= 02 +01vi. = 02vii. Then 01= 02.viii. This statement is a contradiction of what we originally sup-posed.ix. Therefore, there exists one and only one zero vector.x. Q.E.D.Lesson 181. Determine if a subset of a real vector space is itself a vectorspace (i.e., a subspace). Subspace Theorem. If W is a set of oneor more vectors from a vector space V,then W is a subspace of V if and only ifthe following properties hold:(a) Subspace is closed under vector addi-tion:If u and v are vectors in W, thenu +v is also in W.(b) Subspace is closed under scalar multi-plication:If k is any real-valued scalar and u isin W, then ku is also in W.(a) Use the Subspace Theorem to determine which of the followingsubsets of the vector space IR3are subspaces of IR3.linear algebra homework and study guide 37i. S1= (x, 0, 0) : x IRii. S2= (x, 1, 1) : x IRiii. S3= (x, y, z) : y = x + ziv. S4= (x, y, z) : y = x + z +1v. S5= (x, y, 0) : x, y IR(b) Use the Subspace Theorem to determine which of the followingsubsets of the vector space M22 are subspaces of M22.i. T1=__i jk l_ : i, j, k, l ZZ_ii. T2=__w xy z_ : w + x + y + z = 0; w, x, y, z IR_iii. T3=__a bc 0_ : a, b, c IR_iv. T4= A M22 : det (A) = 0v. T5=__x xx x_ : x IR_(c) Use the Subspace Theorem to determine which of the followingsubsets of the vector space P3 are subspaces of P3.i. U1= a0 + a1x + a2x2+ a3x3 P3 : a0= 0ii. U2= a0 + a1x + a2x2+ a3x3 P3 : a0 + a1 + a2 + a3= 0iii. U3= a0 + a1x + a2x2+ a3x3 P3 : a0, a1, a2, a3 0iv. U4= a3x3 P3 : a3 IR(d) Use the Subspace Theorem to determine which of the fol-lowing subsets of the vector space F(, ) are subspaces ofF(, ).i. G1= f F(, ) : f (x) 0, x IRii. G2= f F(, ) : f (0) = 0iii. G3= f F(, ) : f (0) = 2iv. G4= f F(, ) : f (x) = k, k a constantv. G5= f F(, ) : f (x) = k1 +k2sin (x), k1, k2 constants2. Determine if a vector can be represented as a linear combinationof other vectors. A vector w is called a linear combina-tion of the vectors v1, v2, . . . , vn if it canbe expressed in the formw = k1v1 + k2v2 + + knvn=ni=1kiviwhere k1, k2, . . . , kn IR are scalars.(a) If possible, write (3, 3, 6) as a linear combination of x =(1, 2, 1) and y=(0, 1, 1). Otherwise, explain why it can-not be done.(b) Find vectors v and w so that v + w=(4, 5, 6) and v w=(2, 5, 8).linear algebra homework and study guide 38(c) If possible, write_1 57 1_ as a linear combination of A=_4 02 2_, B=_1 12 3_, and C=_0 21 4_. Otherwise, explainwhy it cannot be done.(d) If possible, write 6 +11x +6x2as a linear combination of p1=2 + x +4x2, p2= 1 x +3x2, and p3= 3 +2x +5x2. Otherwise,explain why it cannot be done.3. Determine if a group of vectors span a vector space. Linear Combinations Form a SubspaceTheorem. If v1, v2, . . . , vn are vectors ina vector space V, then the set W of alllinear combinations of v1, v2, . . . , vn is asubspace of V.W= span v1, v2, . . . , vn is a subspace of V.(a) Is IR2= span v, w where v = (2, 3) and w = (1, 4)?(b) Is IR2= span v, w where v = (2, 8) and w = (1, 4)?(c) Is IR2= span v where v = (3, 5)?(d) Is IR3=span x, y, z where x=(2, 2, 2), y=(0, 0, 3), z=(0, 1, 1)?(e) Is IR3=span x, y, z where x =(2, 1, 3), y =(4, 1, 2),z = (8, 1, 8)?(f) Is P2= span p1, p2, p3, p4 where p1= 1 x +2x2, p2= 3 + x,p3= 5 x 4x2, p4= 2 2x +2x2?4. Determine if a vector lies within a space spanned by a set ofvectors.(a) Let f = cos2x and g = sin2x. Which of the following vectors liein span f, g?i. Is cos 2x span f, g?ii. Is 3 + x2 span f, g?iii. Is 1 span f, g?iv. Is 0 span f, g?v. Is sin x span f, g?(b) Let x=(2, 1, 0, 3), y=(3, 1, 5, 2), and z=(1, 0, 2, 1). Whichof the following vectors lie in span x, y, z?i. Is (2, 3, 7, 3) span x, y, z?ii. Is (0, 0, 0, 0) span x, y, z?iii. Is (1, 1, 1, 1) span x, y, z?(c) Describe the geometry of a space spanned by a set of vectors.What geometric shape is represented by spans below? Thengive the shapes mathematical equation or set of equations.i. span (2, 5)ii. span (3, 1), (6, 2)iii. span (2, 1, 3)linear algebra homework and study guide 39iv. span (2, 1), (3, 4)v. span (3, 2, 2), (6, 4, 4)vi. span (1, 1, 1), (4, 4, 3)(d) Describe the solution space (the space of the solution vectors)for a given homogeneous linear system of equations. Solutions to a Homogeneous SystemForm a Subspace Theorem. If Ax= 0is a homogeneous linear system of mequations in n unknowns, then the setof solution vectors is a subspace of IRn.For each homogeneous system of equations below, determine(1) the number of equations m, (2) the number of unknownsn, (3) the solution as a set of parametric equations, and (4) thesolution space of the system Ax=0 given as a set of linearcombinations of vectors.i.x1 2x2+ 3x3 2x4= 03x1+ x2 3x3+ x4= 02x1+ 4x2+ 3x3 x4= 0ii.x3+ x4+ x5= 0x1 x2+ 2x3 3x4+ x5= 0x1+ x2 2x3 x5= 02x1+ 2x2 x3+ x5= 0iii.2x1 x2 3x3= 0x1+ 2x2 3x3= 0x1+ x2+ 4x3= 0linear algebra homework and study guide 40Lesson 19The trivial solution to a system oflinear equations occurs when all ofthe systems unknowns are equal tozero. For example, x1=x2=0 andc1= c2= c3= 0 are trivial solutions tosystems with variables x1, x2 or c1, c2, c3respectively.1. Dene a linearly independent set of vectors and linearly depen-dent set of vectors.Now, lets multiply each vector in aset by an unknown scalar, add theproducts and set the sum equal to thezero vector. Let us call the resultingequation the trivial vector equation orthe zero vector equation:k1v1 + k2v2 + + knvn= 0State from memory the denition of a linearly independent set ofvectors and a linearly dependent set of vectors.2. Determine whether a set of vectors is linearly independent (LI)or linearly dependent (LD) using the denition or theoremsabout linear independence/dependence.A set of vectors is said to be linearlyindependent if its zero vector equa-tion has only the trivial solution, andlinearly dependent otherwise.Some quick terminology: The car-dinality of a set is the numberof vectors in the set; so the car-dinalities of the following setsA, B, C, states in the USA ,(1, 2), (3, 4), (8, 1), (2, 3), (1, 7)are 3, 50, and 5, respectively. We willdene dimension more precisely later,but, for Euclidean spaces IRn, the di-mension is always equal to the numberof components in a vector. For ex-ample, vectors in IR4have the form(a, b, c, d) with four components; so thedimension of IR4is 4. In general, thedimension of IRnis n.Cardinality Exceeds Dimension Theo-rem If the cardinality of set of vectors inIRnexceeds the dimension of IRn, thenthe set of vectors is linearly dependent.In symbolic notation: If S =w1, w2, . . . , wn is a set of p vec-tors in any space IRn, and if p> n, thenS is linearly dependent.(a) The following sets of vectors are taken from vector spaces IR2,IR3, IR4, P2, and M22. In each case determine whether the setof vectors is linearly independent or dependent. Then explainwhy.i. (1, 2, 4), (5, 10, 20)ii. (3, 1), (5, 7), (2, 4)iii. 3 2x + x2, 6 4x +2x2iv.__5 76 0_,_5 76 0__v. (4, 1, 2), (4, 10, 2)vi. (2, 0, 1), (3, 2, 5)(6, 1, 1), (7, 0, 2)vii. (3, 8, 7, 3), (1, 5, 3, 1), (2, 1, 2, 6), (1, 4, 0, 3)viii. (0, 0, 2, 2), (3, 3, 0, 0), (1, 1, 0, 1)ix. 6 x2, 1 + x +4x2(b) For which real values of do the following vectors form a lin-early dependent set in IR3?x1= (, 12, 12), x2= (12, , 12), x3= (12, 12, )3. Establish properties of linear dependent and independent setsusing mathematical proof.(a) Prove: If x1, x2, x3 is a linearly dependent set of vectors in V,and x4 V, then x1, x2, x3, x4 is also linearly dependent.(b) Prove: If x1, x2, x3 is a linearly independent set of vectors in V,then x1, x2, x3, 0 is linearly dependent.Lesson 201. Dene a basis for a real vector space. State from memory the A set of vectors B= x1, x2, . . . , xnfrom a vector space V is said to be abasis for V if: the set B is linearly independent the set B spans V.denition of a basis of real vector space.2. Determine whether a set of vectors form a basis by using thedenition or other properties.linear algebra homework and study guide 41(a) From inspection, explain why the following sets of vectors arenot bases for the indicated vector spaces.i. Is S = (2, 1), (0, 5), (1, 7) a basis for IR2?ii. Is S = (0, 0), (3, 7) a basis for IR2?iii. Is S = (3, 9), (5, 15) a basis for IR2?iv. Is T= (2, 3, 1), (5, 1, 1) a basis for IR3?v. Is U= 1 +2x +3x2, x 2 a basis for P2?vi. Is W=__1 02 5_,_7 32 6_,_4 46 1_,_3 12 0_,_2 31 1__ abasis for M22?(b) Using the denition of a basis or any theorems about bases,determine whether the following sets of vectors are bases forthe indicated vector spaces.i. Is J= (1, 0, 0), (2, 2, 0), (3, 3, 3) a basis for IR3?ii. Is K = (3, 1, 4), (2, 5, 6), (1, 4, 8) a basis for IR3?iii. Is L = 1 3x +2x2, 1 + x +4x2, 1 7x a basis for P2?iv. Is M = 1 + x + x2, x + x2, x2 a basis for P2?v. Is N=__3 63 6_,_0 11 0_,_0 812 4_,_1 01 2__ abasis for M22?3. Calculate the coordinates of vector relative to a specic basis. Uniqueness of Basis RepresentationTheorem. If B= x1, x2, . . . , xn is abasis for a vector space V, then everyvector v V can be written as a linearcombination of ordered basis vectors,v= k1x1 + k2x2 + + knxn, in exactlyone way.(a) If (u)M= (2, 3) and M = (1,3), (23, 1), calculate u.(b) If (v)N=(1, 0, 2) and N= (1, 1, 1), (0, 1, 2), (3, 0, 1),calculate v.(c) If (v)P= (2, 2, 1, 1) andP =__1 00 0_,_0 10 0_,_0 01 0_,_0 00 1__, calculate v.(d) If w=(1, 1), determine (w)S, the coordinate vector of w rela-tive to the basis S = (2, 4), (3, 8).(e) If w=(1, 1), determine (w)T, the coordinate vector of w rela-tive to the basis T= (1, 5), (2, 3).(f) If x=(a, b), determine (x)U, the coordinate vector of x relativeto the basis U= (1, 1), (0, 2).(g) If y=(2, 1, 3), determine (y)V, the coordinate vector of yrelative to the basis V= (1, 0, 0), (2, 2, 0), (3, 3, 3).(h) If u=_2 01 3_, determine (u)J, the coordinate vector of urelative to the basis J=__1 10 0_,_1 10 0_,_0 01 0_,_0 00 1__.linear algebra homework and study guide 424. Determine the basis and dimension for the solution space of ahomogeneous system of linear equations. It can be shown that, if a vector spacehas a basis consisting of n vectors,then any other basis set will alsoconsist of n vectors. The dimensionof a nite-dimensional vector spaceis the number of vectors in any of itsbases. For example, we know thatB= (1, 0, 0), (0, 1, 0), (0, 0, 1) is abasis for IR3so the dimension of IR3is 3(symbolically: dim(IR3)= 3). A basisfor P3 is 1, x, x2, x3, so dim(P3) = 4.For the homogeneous systems of linear equations below, solve thesystem to nd the solution space. Then analyze the solutions tond the basis and the dimension of the solution space.(a)3x1+ x2+ 4x3 x4= 02x1 x2+ x3 x4= 0(b)x + y + z = 03x + 2y 2z = 04x + 3y z = 06x + 5y + 2z = 0(c)x1 3x2+ x3= 02x1 6x2+ 2x3= 03x1 9x2+ 3x3= 0(d)x1+ 3x2 x3= 02x1 5x3= 0x2+ 2x3= 05. Determine the basis and dimension of subspaces describedalgebraically and geometrically .(a) Find the basis and dimension of the following subspaces of IR3.i. the plane 2x 3y +7z = 0ii. the plane x 3y = 0iii. the line x = 2t, y = 3t, z = 5tiv. Is the origin__000__ included in the subspaces of the three exer-cises above? Explain.v. (x, y, z) IR3: y = x + z(b) Find the basis and dimension of the following subspaces of IR4.i. (w, x, y, 0) : w, x, y IRii. (w, x, y, z) IR4: w = x = y = ziii. (w, x, y, z) IR4: y = w x, z = w + x(c) Find the basis and dimension of the following subspace of P4:k4x4+ k3x3+ k2x2+ k1x + k0: k0= 0.Lesson 211. Identify row and column vectors for a given matrix.linear algebra homework and study guide 43List the row and column vectors of the matrix A.__3 2 5 4 62 1 5 1 04 2 3 1 2__2. Express matrix-vector products (e.g. Ax) as linear combinationsof column vectors of the matrix.Express the matrix-vector product Ax as a linear combination ofthe column vectors of A.(a)_2 51 3_ _34_(b)__3 0 11 3 25 1 2____142__(c)__3 6 14 5 12 1 03 1 1____125__3. Determine whether a given column vector lies in the columnspace of a matrix. Column Space Test for ConsistencyTheorem. A system of linear equationsAx= b is consistent if and only if b isin the column space of A. In symbols,Ax = b is consistent b ( (A)Determine whether the column vector b is in the column spaceof A. If so, then express b as a linear combination of the columnvectors of A.(a) A =_1 34 6_; b =_210_(b) A =__1 1 21 0 12 1 3__; b =__102__(c) A =__1 1 19 3 11 1 1__; b =__511__(d) A =__1 1 11 1 11 1 1__; b =__200__(e) A =__1 2 0 10 1 2 11 2 1 30 1 2 2__; b =__2357__linear algebra homework and study guide 444. Explain the distinctions and the relationships between the termparticular solution for a nonhomogeneous linear system Ax = b,the general solution for a homogeneous linear system Ax=0,and the general solution of the nonhomogeneous linear systemAx = b.(a) Suppose that x=__1253__+__1253__s +__125237__t solves thesystem Ax = b.i. What is the particular solution of Ax = b?ii. What is the general solution of Ax = 0?iii. What is the general solution of Ax = b?(b) Suppose that x1= 9, x2= 3, x3= 1, x4= 34is a solution of anonhomogeneous linear system Ax = b and that the solution ofthe homogeneous system Ax = 0 is given by x1= 2 3, x2= 2, x3= , x4=.i. Find the vector form of the general solution of Ax = 0.ii. Find the vector form of the general solution of Ax = b.(c) For each of the problem below, rst nd the vector form of thegeneral solution of Ax=b; then nd the vector form of thegeneral solution of Ax = 0.i.3x1 x2= 16x1+ 2x2= 2ii.x1+ x2+ 2x3= 5x1+ x3= 22x1+ x2+ 3x3= 3iii.x1 2x2+ x3+ 2x4= 12x1 4x2+ 2x3+ 4x4= 2x1 2x2 x3 2x4= 13x1 6x2+ 3x3+ 6x4= 35. Find the bases for row space and null space of a matrix that is inrow-echelon form.The matrices Ri below are all in row-echelon form. Find bases for!(Ri), row space of Ri, and ( (Ri), column space of Ri.(a) R1=__1 0 30 0 10 0 0__linear algebra homework and study guide 45(b) R2=__1 5 0 00 1 0 00 0 0 00 0 0 0__(c) R3=__1 3 5 40 1 3 00 0 1 40 0 0 10 0 0 0__(d) R4=__1 2 1 60 1 5 20 0 1 90 0 0 1__6. Given a matrix that is not in row-echelon form, nd the basesfor row space, column space, and nullspace of the given matrix.Find bases for !(Ai), row space of Ai, ( (Ai), column space of Ai,and A(Ai), null space of Ai.(a) A1=__1 1 35 4 47 6 2__(b) A2=__1 4 5 22 1 3 01 3 2 2__(c) A3=__1 4 5 6 93 2 1 4 11 0 1 2 12 3 5 7 8__(d) A4=__1 3 2 2 10 3 6 0 32 3 2 4 43 6 0 6 52 9 2 4 5__7. Find a basis for row space of a matrix consisting entirely of rowvectors of the given matrix.For the matrices below, nd bases for !(A) that consist entirely ofrow vectors of A.(a) A =__1 2 0 0 32 5 3 2 60 5 15 10 02 6 18 8 6__linear algebra homework and study guide 46(b) A =__1 3 4 2 5 42 6 9 1 8 22 6 9 1 9 71 3 4 2 5 4__8. Construct a matrix that generates a particular nullspace.(a) Let A =__0 1 01 0 00 0 0__.Show that A(A) is the z-axis and that ( (A) is the xy-plane.(b) Find a 33 matrix whose nullspace is the x-axis and whosecolumn space is the yz-plane.(c) Find a 3 3 matrix whose nullspace is:i. a pointii. a lineiii. a plane9. Find a basis for a subspace that is the span of a set of vectors.(a) For each subspace of IR4below, nd a basis.i. S1= span (1, 2, 4, 3), (2, 0, 5, 2), (2, 2, 1, 2)ii. S2= span (1, 1, 2, 1), (3, 3, 6, 0), (7, 1, 14, 4)iii. S3= span (2, 2, 0, 0), (2, 1, 2, 5), (2, 0, 2, 2), (0, 3, 0, 3)(b) For each subspace of IR4below, nd a basis in terms of thevectors of the spanning set. Then express each vector not a partof the basis as a linear combination of the remaining vectors.i. T1= span (1, 0, 1, 1), (3, 3, 7, 1), (1, 3, 9, 3), (5, 3, 5, 1)ii. T2= span (1, 2, 0, 3), (2, 4, 0, 6), (1, 1, 2, 0), (0, 1, 2, 3)linear algebra homework and study guide 47Lesson 221. Dene the four fundamental matrix spaces. Denition of the Four FundamentalMatrix Spaces. Three vector spaces canbe generated for a matrix A, namely,rowspace of A, columnspace of A,and nullspace of A, and three vectorspaces can be generated from AT,namely, rowspace of AT, columnspaceof AT, and nullspace of AT. Note,however, that !(A) = ( (AT) and( (A) = !(AT). So, any matrixA is said to generate a total of fourfundamental matrix spaces: !(A) ( (A) A(A) A(AT)State from memory the four fundamental matrix spaces.2. Dene the rank of a matrix.Denition of Rank of a Matrix. Therank of a matrix A is the dimensionof !(A) (or ( (A)). Note that thedimension of !(A) and ( (A) arealways the same. The rank of A isdenoted by rank (A).(a) State from memory the denition of the rank of a matrix.(b) Explain why rank (A) = rank (AT).3. Dene the nullity of a matrix.Denition of Nullity of a Matrix. Thenullity of a matrix A is the dimensionof A(A). The nullity of A is denotedby nullity (A).State from memory the denition of the nullity of a matrix.4. Using the Dimension Theorem, explain the relationship betweenthe rank of a given matrix, its nullity, and the number of columnvectors in the matrix.Dimension Theorem for Matrices IfA is a matrix with n columns, thenrank (A) +nullity (A) = n.Give the relationship between rank, nullity, and number of columnsof a matrix in a formula from memory.5. Determine the rank and nullity of a given matrix.Find the rank and nullity of the matrices below. Then verify thatthe rank, nullity, and number of columns satisfy the relationshipdescribed in the Dimension Theorem.(a) Amn=__1 1 35 4 47 6 2__;rank (A) = ; nullity (A) = ; n = .(b) Brs=__1 4 5 22 1 3 01 3 2 2__;rank (B) = ; nullity (B) = ; s = .(c) Cpq=__1 3 2 2 10 3 6 0 32 3 2 4 43 6 0 6 52 9 2 4 5__;rank (C) = ; nullity (C) = ; q = .6. Determine the number of leading (basic) variables and the num-ber of parameters (free variables) for a homogeneous systemAx = 0 from the rank and nullity of the coefcient matrix A.Determine the number of leading variables and the number ofparameters for the homogeneous systems given below. (Note:you have already computed the rank and nullity of these matrices in theprevious problem.)(a) Ax = 0; Amn=__1 1 35 4 47 6 2__linear algebra homework and study guide 48# of leading variables = ; # of parameters = .(b) Bx = 0; Brs=__1 4 5 22 1 3 01 3 2 2__# of leading variables = ; # of parameters = .(c) Cx = 0; Cpq=__1 3 2 2 10 3 6 0 32 3 2 4 43 6 0 6 52 9 2 4 5__# of leading variables = ; # of parameters = .7. Determine the largest possible rank and smallest possible nul-lity given the size of a matrix.Find the largest possible rank and the smallest possible nullity forthe following matrices:(a) S44(b) T35(c) U53(d) Amn8. Determine the dimensions of the four fundamental subspacesof a matrix given the size of the matrix and its rank. Find the Dimensions of Fundamental MatrixSpaces Theorem. If a matrix Apqhas rank r, then the dimensions of thefour fundamental matrix spaces are asfollows: dim!(A) = r dim( (A) = r dimA(A) = q r dimA(AT) = p rdimension of row space, column space, and nullspace of the givenmatrix as well as the dimension of nullspace of the transpose ofthe matrix.(a) If rank (A33) = 3, then (i) dim(!(A)) = ;(ii) dim(( (A)) = ; (iii) dim(A(A)) = ; and(iv) dim(A(AT)) = .(b) If rank (A33) = 2, then (i) dim(!(A)) = ;(ii) dim(( (A)) = ; (iii) dim(A(A)) = ; and(iv) dim(A(AT)) = .(c) If rank (A33) = 1, then (i) dim(!(A)) = ;(ii) dim(( (A)) = ; (iii) dim(A(A)) = ; and(iv) dim(A(AT)) = .(d) If rank (B59) = 2, then (i) dim(!(B)) = ;(ii) dim(( (B)) = ; (iii) dim(A(B)) = ; and(iv) dim(A(BT)) = .(e) If rank (C95) = 2, then (i) dim(!(C)) = ;(ii) dim(( (C)) = ; (iii) dim(A(C)) = ; and(iv) dim(A(CT)) = .linear algebra homework and study guide 49(f) If rank (D44) = 0, then (i) dim(!(D)) = ;(ii) dim(( (D)) = ; (iii) dim(A(D)) = ; and(iv) dim(A(DT)) = .(g) If rank (E62) = 2, then (i) dim(!(E)) = ;(ii) dim(( (E)) = ; (iii) dim(A(E)) = ; and(iv) dim(A(ET)) = .9. Determine whether a system Ax= b is consistent by evaluatingthe rank of the coefcient matrix A and the augmented matrix[A[b]. Additionally use the size of the coefcient matrix A todetermine the number of parameters in the general solution. Augmented Matrix Test for Consis-tency. A system of linear equationsAx=b is consistent if and only ifthe rank of A and the rank of the aug-mented matrix [A[b] are equal. Insymbols, Ax=b is consistent rank (A) = rank ([A[b])Given the size and rank of the coefcient matrix A and given therank of the augmented matrix [A[b], determine whether the asso-ciated linear system Ax= b is consistent. If Ax= b is consistent,determine the number of parameters in its general solution.(a) rank (A33) = 3 and rank ([A[b]) = 3(b) rank (A33) = 2 and rank ([A[b]) = 3(c) rank (A33) = 1 and rank ([A[b]) = 1(d) rank (A59) = 2 and rank ([A[b]) = 2(e) rank (A59) = 2 and rank ([A[b]) = 3(f) rank (A44) = 0 and rank ([A[b]) = 0(g) rank (A62) = 0 and rank ([A[b]) = 2linear algebra homework and study guide 50Test IV: Inner Product SpacesLesson 231. Dene an inner product for a vector space. An inner product on a real vector spaceV is a function that associates a realnumber x, y with each pair of vectorsx and y in V such that the followingproperties are satised for all vectorsx, y, and z in V and all real numberscalars k:(a) Symmetry:x, y = y, x(b) Additivity:x +y, z = x, z +y, z(c) Homogeneity:kx, y = k x, y(d) Positivity:i. x, x 0ii. x, x = 0x = 0State from memory the four properties of the denition of an innerproduct.2. Determine whether a particular function that maps two vectorsto a scalar is or is not an inner product.Are the following mappings inner products on the given space? Ifso, show that all four properties of the inner product function aresatised. If not, show that at least one of the properties does nothold.(a) Let x = (x1, x2) and y = (y1, y2) be any vectors in IR2.x, y = 2x1y1 +7x2y2(b) Let x = (x1, x2) and y = (y1, y2) be any vectors in IR2.x, y = x1y1x2y2(c) Let u = (u1, u2, u3) and v = (v1, v2, v3) be any vectors in IR3.u, v = u1v1 + u3v3(d) Let u = (u1, u2, u3) and v = (v1, v2, v3) be any vectors in IR3.u, v = 2u1v1 + u2v2 +4u3v3(e) Let u = (u1, u2, u3) and v = (v1, v2, v3) be any vectors in IR3.u, v = u1v1u2v2 + u3v33. Give examples of how different inner products yield differentvalues for magnitude of a vector and distance between vectors.(a) If the inner product on IR2is the Euclidean inner productx, y = x1y1 + x2y2, nd:i. [[w[[ if w = (2, 5)ii. d(a, b) if a = (1, 2) and b = (2, 5)(b) If the inner product on IR2is the weighted Euclidean innerproduct x, y = 2x1y1 +3x2y2, nd:i. [[w[[ if w = (2, 5)ii. d(a, b) if a = (1, 2) and b = (2, 5)(c) If the inner product x, y on IR2is generated by the matrix_1 21 3_, nd:i. [[w[[ if w = (2, 5)ii. d(a, b) if a = (1, 2) and b = (2, 5)(d) If p= a0 + a1x + a2x2and q= b0 + b1x + b2x2are vectors in P2and p, q = a0b0 + a1b1 + a2b2, ndlinear algebra homework and study guide 51i. [[r[[ if r = 2 +3x + x2ii. d(r, s) is r = 5 2x2and s = x +2x2(e) If p=p(x)=a0 + a1x + a2x2and q= q(x)= b0 + b1x + b2x2are vectors in P2 and p, q =_11p(x)q(x)dx, ndi. [[r[[ if r = x2ii. d(r, s) if r = 2 x and s = 1 +2x x2(f) If =_a11a12a21a22_ and=_b11b12b21b22_ are vectors in M22 and, = a11b11 + a12b12 + a21b21 + a22b22, ndi. [[[[ if =_2 67 3_ii. d(, ) if =_4 71 5_ and =_2 41 0_4. Evaluate specic inner products using their properties. Supposethat x, y, and z are vectors in space V such thatx, y=3, y, z= 2, x, z=4, [[x[[ =1, [[y[[ =2, [[z[[ =5.Evaluate the following expressions.(a) x +y, y +z(b) 3y +z, 2x +2z(c) [[x +z[[(d) [[2y z[[5. Prove theorems about inner products using the denitions ax-ioms.(a) Prove: If u and v are vectors in a real inner product space V andk IR, then u, kv = k u, v.(Justify each step of the proof below with the appropriate math-ematical property, theorem, or denition.)i. Proof. Assumeii. u, kv = kv, uiii. = k v, uiv. = k u, vv. Therefore, u, kv = k u, vvi. Q.E.D.(b) Prove: If u, v, and w are vectors in a real inner product space Vand k IR, then u v, w = u, w v, w.(c) Prove: If u, v, and w are vectors in a real inner product space Vand k IR, then u, v w = u, v u, w.linear algebra homework and study guide 52Lesson 241. Determine if two vectors are orthogonal using a given innerproduct.Find the angle between two vectors for a given inner product.(a) Determine if, or under what circumstances, the pair of vectorsbelow are orthogonal under the Euclidean inner product.i. u = (1, 3, 2) and v = (4, 2, 1)ii. x = (3, 3, 3) and y = (1, 0, 1)iii. a = (a1, a2, a3) and b = (0, 0, 0)iv. = (0, 3, 2, 1) and = (5, 2, 1, 0)v. r = (2, 1, 3) and s = (1, 7, c)vi. v = (k, k, 1) and w = (k, 5, 6)(b) Do there exist scalars a and b such that vectors x=(2, a, 6),y=(b, 5, 3), and z=(1, 2, 3) are pairwise orthogonal withrespect to the Euclidean inner product?(c) Solve the following equation for k assuming that IR3has theEuclidean inner product and vectors u=(1, 1, 1) and v=(6, 7, 15):[[ku +v[[ = 13(d) Let p, q= a0b0 + a1b1 + a2b2 be the inner product for any twovectors p = a0 + a1x + a2x2and q = b0 + b1x + b2x2in P2.i. Determine whether the two vectors p=1 x + 2x2andq = 2x + x2are orthogonal.ii. Find the angle between the two vectors p= 1 + 5x + 2x2and q = 2 +4x 3x2(e) Let p, q=_11p(x)q(x)dx be the inner product for any twovectors p = a0 + a1x + a2x2and q = b0 + b1x + b2x2in P2.i. Determine whether the two vectors p=1 x + 2x2andq = 2x + x2are orthogonal.ii. Find the angle between the two vectors p= 1 + 5x + 2x2and q = 2 +4x 9x2(f) Let , =a11b11 + a12b12 + a21b21 + a22b22 be the innerproduct for any two vectors =_a11a12a21a22_ and =_b11b12b21b22_in M22.i. Determine whether the two vectors =_2 11 3_ and =_3 00 2_ are orthogonal.linear algebra homework and study guide 53ii. Find the angle between the two vectors =_1 42 3_ and =_3 20 1_2. Given a subspace W of a vector space V, nd its complement.(a) Let W1 be the line in IR2with equation y = 2x. Find an equationfor W1.(b) Let W2 be the plane in IR3with equation x 2y 3z= 0. Findparametrics for W2.(c) Let W3 be the line in IR3with parametric equations x =2t,y = 5t, z = 4t. Find an equation for W3.(d) Let W4 be the intersection of the two planes x + y + z= 0 andx y + z = 0. Find an equation for W4.3. Explain the geometric link between nullspace of A and the rowspace of A. The Fundamental Theorem of LinearAlgebra. If A be an mn matrixwith rank r, then four fundamentalsubspaces can be formed: !(A), the row space of A; ( (A), the column space of A; A(A). the nullspace of A; and A(AT), the nullspace of AT(1) And the dimensions of the fourfundamental subspaces are:dim!(A) = dim( (A) = r,dimA(A) = n r,dimA(AT) = mr(2) And the geometric relationships be-tween the subspaces are as follows:i. A(A) = (!(A))The nullspace of A and therow space of A are orthogonalcomplements in IRnwith respectto the Euclidean inner product.ii. A(AT) = (( (A))The nullspace of ATand the col-umn space of A are orthogonalcomplements in IRmwith respectto the Euclidean inner product.LetA =__1 2 1 23 5 0 41 1 2 0__(a) Find bases for !(A) and A(A).(b) Verify that every vector in !(A) is orthogonal to every vectorin A(A) as guaranteed by the Fundamental Theorem of LinearAlgebra.4. Explain the geometric link between nullspace and column spaceof A. LetA =__1 2 1 23 5 0 41 1 2 0__(a) Find bases for ( (A) and A(AT).(b) Verify that every vector in ( (A) is orthogonal to every vector inA(AT) as guaranteed by the Fundamental Theorem of LinearAlgebra.Lesson 251. Dene an orthogonal set of vectors and an orthonormal set ofvectors. A set of vectors in an inner productspace is said to be an orthogonal setif all pairs of vectors in the set areorthogonal. An orthonormal set is anorthogonal set in which the each vectorhas length 1.(a) State from memory the denition of an orthogonal set of vec-tors.linear algebra homework and study guide 54(b) State from memory the denition of an orthonormal set ofvectors.2. Classify a set of vectors orthonormal, orthogonal only, or notorthogonal.(a) For each set of vectors below, determine whether the set ofvectors is orthonormal, orthogonal only, or not orthogonal withrespect to the Euclidean inner product.i. (0, 1), (2, 0)ii. _12,12_,_12,12_iii. _12, 12_,_12,12_iv. (1, 0), (0, 0)v. _12, 0,12_,_13,13, 13_,_12, 0,12_vi. _23, 23, 13_,_23, 13, 23_,_13, 23, 23_vii. (1, 0, 0) ,_0,12,12_, (0, 0, 1)viii. _16,16, 26_,_12, 12, 0_(b) For the set of vectors below, determine whether the set of vec-tors is orthonormal, orthogonal only, or not orthogonal withrespect to the inner product p, q=a0b0 + a1b1 + a2b2, wherep = a0 + a1x + a2x2and q = b0 + b1x + b2x2in P2.i._23 23x + 13x2, 23+ 13x 23x2, 13+ 23x + 23x2_ii._1,12x +12x2, x2_(c) For the set of vectors below, determine whether the set of vec-tors is orthonormal, orthogonal only, or not orthogonal withrespect to the inner product , =a11b11 + a12b12 + a21b21 +a22b22, where =_a11a12a21a22_ and =_b11b12b21b22_ in M22.i.__1 00 0_,_0231323_,_0232313_,_0132323__ii.__1 00 0_,_0 10 0_,_0 01 1_,_0 01 1__3. Given an orthonormal basis, use the Coordinate Vector Theo-rem to write a vector as a linear combination of vectors in theorthonormal set. The Coordinate Vector Theorem. IfB= b1, b2, . . . , bn is an orthonormalbasis for an inner product space V andv is any vector in V, thenv = v, b1 b1 +v, b2 b2 + +v, bn bnAnd we can write v as a coordinatevector (v)B:(v)B= (v, b1 , v, b2 , . . . , v, bn)linear algebra homework and study guide 55(a) Verify that the vectors x=(35, 45, 0), y=(45, 35, 0), z=(0, 0, 1)form an orthonormal basis for IR3with the Euclidean innerproduct. Then use the Coordinate Vector Theorem to expresseach of the following as linear combinations of x, y, and z.i. u = (1, 1, 2)ii. v = (3, 7, 4)iii. w = (17, 37, 57)(b) Using the information from the previous problem, rewrite eachvector as a coordinate vector with respect to the orthonormalbasis B = x, y, z.i. (u)B=ii. (v)B=iii. (w)B=(c) Find the coordinate vector with respect to the given orthonor-mal basis. (In both cases, assume the Euclidean inner product.)i. Find (r)B where r =(3, 7) and B=b1, b2, in whichb1=_12, 12_ and b2=_12,12_ii. Find (s)B where s =(1, 0, 2) and B= b1, b2, b3, in whichb1=_23, 23, 13_, b2=_23, 13, 23_, and b3=_13, 23, 23_4. Compute magnitudes, distances, and inner products using co-ordinates as allowed by the Coordinate-Magnitude Properties.Coordinate-Magnitude PropertiesTheorem. If B is an orthonormal basisfor an n-dimensional inner productspace V and if the coordinate vectorsrelative to B for two vectors x and y aregiven by(x)B= (x1, x2, . . . , xn) and (y)B= (y1, y2, . . . , yn)theni. x, y = x1y1 + x2y2 + + xnynii. [[x[[ =_x21 + x22 + + x2niii. d(x, y) =_(x1 y1)2+ + (xn yn)2(a) Let IR2have the Euclidean inner product, and let B= b1, b2be an orthonormal basis with b1=_35, 45_, b2=_45, 35_i. Find vectors x and y in IR2that have coordinate vectors(x)B= (1, 1) and (y)B= (1, 4).ii. Compute x, y, [[x[[, and d(x, y) by applying the Coordinate-Magnitude Properties to the coordinate vectors (x)B and(y)B; then recalculate the inner product, the norm, and thedistance between two vectors by performing the necessarycomputations directly on x and y.(b) Let IR3have the Euclidean inner product, and let B = b1, b2, b3be an orthonormal basis with b1=_0, 35, 45_, b2=(1, 0, 0),b3=_0, 45, 35_i. Find vectors x, y, and z in IR3that have coordinate vectors(x)B= (2, 1, 2), (y)B= (3, 0, 2), and (z)B= (5, 4, 1).linear algebra homework and study guide 56ii. Compute x, z, [[y[[, and d(z, y) by applying the Coordinate-Magnitude Properties to the coordinate vectors (x)B and(y)B; then recalculate the results by performing the necessarycomputations directly on x, y, and z.5. Use the Gram-Schmidt process to transform a regular basis intoan orthonormal one.(a) Let IR2have the Euclidean inner product. Perform the Gram-Schmidt process on each of the bases for IR2given below to ndan associated orthogonal basis. Then divide out the magnitudeof each vector in the orthogonal basis to nd an orthonormalone.i. (1, 1), (2, 1)ii. (0, 1), (1, 3)iii. (0, 1), (1, 0)(b) Let IR3have the Euclidean inner product. Perform the Gram-Schmidt process on each of the bases for IR3given below to ndan associated orthogonal basis. Then divide out the magnitudeof each vector in the orthogonal basis to nd an orthonormalone.i. (2, 2, 2), (1, 0, 1), (0, 3, 1)ii. (1, 1, 0), (0, 1, 0), (2, 3, 1)(c) Note that the plane x y + z= 0 is a subspace of IR3. Find anorthonormal basis for it.(d) Note that the set(x, y, z, w) IR4: x y z + w = 0 and x + z = 0is a subspace of IR4. Find an orthonormal basis for it.Lesson 261. Given an inconsistent linear system, nd the best approximationto the system (i.e., the least squares solution).Given a least squares solution to an inconsistent linear system,nd the orthogonal projection of it onto the column space of thecoefcient matrix of the original system.For each inconsistent system Ax= b, give the best approximation(least squares solution). Then nd the orthogonal projection of bonto the column space of A.(a)x + y = 7x + y = 0x + 2y = 7linear algebra homework and study guide 57(b)2x 2y = 2x + y = 13x + y = 1(c)x z = 62x + y 2z = 0x + y = 9x + y z = 3(d)2x z = 0x 2y + 2z = 62x y = 0y z = 6(e) Compute the orth