# linear algebra

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• 1. Chapter 1 Systems of Linear Equations Linear Algebra

2. Ch1_2 Definition An equation such as x+3y=9 is called a linear equation. ( ) The graph of this equation is a straight line in the x-y plane. A pair of values of x and y that satisfy the equation is called a solution. 1.1 Matrices and Systems of Linear Equations system of linear equations ( ) 3. Ch1_3 Definition A linear equation in n variables ( ) x1, x2, x3, , xn has the form a1 x1 + a2 x2 + a3 x3 + + an xn = b where the coefficients ( ) a1, a2, a3, , an and b are real numbers ( ). natural number ( ), integer ( ), rational number ( ), real number ( ), complex number ( ) positive ( ), negative ( ) 4. Ch1_4 Figure 1.2 No solution ( ) 2x + y = 3 4x + 2y = 2 Lines are parallel. No point of intersection. No solutions. Solutions for system of linear equations Figure 1.1 Unique solution ( ) x + 3y = 9 2x + y = 4 Lines intersect at (3, 2) Unique solution: x = 3, y = 2. Figure 1.3 Many solution ( ) 4x 2y = 6 6x 3y = 9 Both equations have the same graph. Any point on the graph is a solution. Many solutions. 5. Ch1_5 A linear equation in three variables corresponds to a plane in three-dimensional ( ) space. Unique solution Systems of three linear equations in three variables: 6. Ch1_6 No solutions Many solutions 7. Ch1_7 How to solve a system of linear equations? Gauss-Jordan elimination. ( - ) 1.2 8. Ch1_8 Definition A matrix ( ) is a rectangular array of numbers. The numbers in the array are called the elements ( ) of the matrix. Matrices = = = 1298 520 653 C 38 50 17 B 157 432 A 9. Ch1_9 Submatrix ( ) Amatrix 215 032 471 =A Row ( ) and Column ( ) [ ] [ ] 3column2column1column2row1row 1 4 5 3 7 2 157432 157 432 =A Aofssubmatrice 25 41 1 3 7 15 32 71 = = = RQP 10. Ch1_10 Identity Matrices ( ) diagonal ( ) 1 0 I size = = 100 010 001 10 01 32 II Location 7,4 157 432 2113 == = aaA aij row i, column j location (1,3) = 4 Size and Type [ ] matrixcolumnamatrixrowamatrixsquarea matrix13matrix41matrix3332:Size 2 3 8 5834 853 109 752 542 301 11. Ch1_11 matrix of coefficient and augmented matrix 62 332 2 321 321 321 = =++ =++ xxx xxx xxx Relations between system of linear equations and matrices 5(f) tcoefficienofmatrix 211 132 111 matrixaugmented 6211 3132 2111 12. Ch1_12 Elementary Transformation 1. Interchange two equations. 2. Multiply both sides of an equation by a nonzero constant. 3. Add a multiple of one equation to another equation. Elementary Row Operation ( ) 1. Interchange two rows of a matrix. ( ) 2. Multiply the elements of a row by a nonzero constant. ( ) 3. Add a multiple of the elements of one row to the corresponding elements of another row. ( ) Elementary Row Operations of Matrices 13. Ch1_13 Example 1 Solving the following system of linear equation. 62 332 2 321 321 321 = =++ =++ xxx xxx xxx 6211 3132 2111 62 332 2 321 321 321 = =++ =++ xxx xxx xxx Solution Equation Method Initial system: Analogous Matrix Method Augmented matrix: 1 2 32 321 = =++ xx xxx Eq2+(2)Eq1 Eq3+(1)Eq1 8320 1110 2111 R2+(2)R1 R3+(1)R1 row equivalent 832 32 = xx 14. Ch1_14 105 1 32 3 32 31 = = =+ x xx xx 01500 1110 3201 2 1 32 3 32 31 = = =+ x xx xx 2100 1110 3201 2 1 1 3 2 1 = = = x x x 2100 1010 1001 Eq1+(1)Eq2 Eq3+(2)Eq2 (1/5)Eq3 Eq1+(2)Eq3 Eq2+Eq3 The solution is .2,1,1 321 === xxx The solution is .2,1,1 321 === xxx 832 1 2 32 32 321 = = =++ xx xx xxx 8320 1110 2111 R1+(1)R2 R3+(2)R2 (1/5)R3 R1+(2)R3 R2+R3 7(d) 15. Ch1_15 Example 2 Solving the following system of linear equation. 833 1852 1242 321 321 321 =+ =+ =+ xxx xxx xxx Solution ( ) 8331 18512 12421 4110 6330 12421 R2 3 1 4110 2110 12421 6200 2110 8201 3100 2110 8201 R1R3 2)R1(R2 + + R2)1(R3 (2)R2R1 + + R3 2 1 3100 1010 2001 R3R2 2)R3(R1 + + . 3 1 2 solution 3 2 1 = = = x x x 16. Ch1_16 Example 3 Solve the system 72 32863 441284 21 321 321 = =+ =+ xx xxx xxx 7012 32863 441284 7012 32863 11321 15630 1100 11321 1100 5210 11321 1100 5210 1101 . 1100 3010 2001 .1,3,2issolutionThe 321 === xxx R2 3 1 R12R3 3)R1(R2 + + R3R2 1100 15630 11321 R1 4 1 2)R2(R1 + 2R3R2 1)R3(R1 + + Solution ( ) 10(d)(f) 17. Ch1_17 Summary = 7012 32863 441284 ]:[ BA A BUse row operations to [A: B] : . 1100 3010 2001 7012 32863 441284 ]:[]:[ XIBA ni.e., Def. [In : X] is called the reduced echelon form ( ) of [A : B].Note. 1. If A is the matrix of coefficients of a system of n equations in n variables that has a unique solution, then A is row equivalent to In (A In). 2. If A In, then the system has unique solution. 72 32863 441284 21 321 321 = =+ =+ xx xxx xxx 18. Ch1_18 Example 4 Many Systems Solving the following three systems of linear equation, all of which have the same matrix of coefficients. 3321 2321 1321 42 for42 3 bxxx bxxx bxxx =+ =+ =+ in turn 4 3 3 , 2 1 0 , 11 11 8 3 2 1 = b b b Solution 4211421 3111412 308311 . 2 1 2 , 1 3 0 , 2 1 1 3 2 1 3 2 1 3 2 1 = = = = = = = = = x x x x x x x x x 123110 315210 308311 212100 315210 013101 212100 131010 201001 R2+(2)R1 R3+R1 1)R2(R3 R2R1 + + R32R2 R3)1(R1 + + 13(b) The solutions to the three systems are 19. Ch1_19 Homework Exercise 1.1 1, 2, 4, 5, 6, 7, 10, 13 20. Ch1_20 1-2 Gauss-Jordan Elimination Definition A matrix is in reduced echelon form ( ) if 1. Any rows consisting entirely of zeros are grouped at the bottom of the matrix. 2. The first nonzero element of each other row is 1. This element is called a leading 1. 3. The leading 1 of each row after the first is positioned to the right of the leading 1 of the previous row. 4. All other elements in a column that contains a leading 1 are zero. 21. Ch1_21 Examples for reduced echelon form 10000 04300 03021 9100 3010 7001 3100 0000 4021 000 210 801 () ()() () elementary row operations reduced echelon form The reduced echelon form of a matrix is unique. 2(b)(d) (h) 22. Ch1_22 Gauss-Jordan Elimination System of linear equations augmented matrix reduced echelon from solution 23. Ch1_23 + 41200 22200 43111 4)R1(R3 + + 61000 11100 52011 R2)2(R3 R2R1 Example 1 Use the method of Gauss-Jordan elimination to find reduced echelon form of the following matrix. 1211244 129333 22200 Solution 1211244 22200 129333 R2R1 pivot ( leading 1) 1211244 22200 43111 R1 3 1 pivot + + 61000 50100 170011 R3R2 2)R3(R1 The matrix is the reduced echelon form of the given matrix. 41200 11100 43111 R2 2 1 24. Ch1_24 Example 2 Solve, if possible, the system of equations 753 742 9333 321 321 321 = =+ =+ xxx xxx xxx Solution ( ) ( ) 7153 7412 3111 7153 7412 9333 R1 3 1 + + 2420 1210 3111 3)R1(R3 2)R1(R2 + + 0000 1210 4301 R2R3 R2R1 12 43 12 43 32 31 32 31 += += =+ =+ xx xx xx xx The general solution to the system is .parameter)a(callednumberrealiswhere, 12 43 3 2 1 rrx rx rx = += += 5(c) 25. Ch1_25 Example 3 Solve the system of equations 446942 14232 58101242 4321 4321 4321 =+ =++ =+ xxxx xxxx xxxx Solution( ) ( ) 446942 142321 295621 446942 142321 58101242 R1 2 1 + + 144300 153300 295621 2)R1(R3 R1R2 ( ) 144300 51100 295621 R2 3 1 + + 11000 51100 11021 3R2R3 6)R2(R1 + + 11000 60100 20021 R3R2 1)R3(R1 .somefor, 1 6 22 1 6 22 4 3 2 1 4 3 21 r x x rx rx x x xx = = = = = = = > many sol. 26. Ch1_26 Example 4 Solve the system of equations 432 636242 232 54321 54321 54321 =++ =+++ =+++ xxxxx xxxxx xxxxx Solution( ) + + 642000 210000 213121 431121 636242 213121 R1R3 2)R1(R2 210000 642000 213121 R3R2 ( ) 210000 321000 213121 2R 2 1 + 210000 321000 750121 3)R2(R1 + + 210000 101000 300121 2)R3(R2 5R3R1 .andsomefor, 2 ,1,, 32 2 1 32 5 432 1 5 4 321 sr x xsxrx srx x x xxx = === ++= = = ++= 27. Ch1_27 Example 5 This example illustrates a system that has no solution. Let us try to solve the system 122 32 4522 32 32 321 321 321 =+ =+ =+ =+ xx xxx xxx xxx Solution( ) 1220 2100 6330 3211 R3R2 ( ) 1220 2100 2110 3211 R2 3 1 + + 5400 2100 2110 1101 2)R2(R4 R2R1 + + + 13000 2100 4010 3001 4)R3R(R4 R3R2 1)R3(R1 ( ) 1000 2100 4010 3001 R4 13 1 The system has no solution. + + 1220 6330 2100 3211 1220 3121 4522 3211 1)R1(R3 2)R1(R2 0x1+0x2+0x3=1 5(d) 28. Ch1_28 Homogeneous System of linear Equations Definition A system of linear equations is said to be homogeneous ( ) if all the constant terms ( ) are zeros. Example: =+ =+ 0632 052 321 321 xxx xxx Observe that is a solution.0,0,0 321 === xxx Theorem 1.1 A system of homogeneous linear equations in n variables always has the solution x1 = 0, x2 = 0. , xn = 0. This solution is called the trivial solution. 29. Ch1_29 Homogeneous System of linear Equations Theorem 1.2 A system of homogeneous linear equations that has more variables than equations has many solutions. Note. trivial solution 8(e) 0410 0301 0632 0521 =+ =+ 0632 052 321 321 xxx xxx The system has other nontrivial solutions. rxrxrx === 321 ,4,3 Example: 30. Ch1_30 Homework Exercise 1.2: 2, 5, 8, 14 (Section 1.3 )

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