Department of Mathematics, University of TorontoMAT223H1S - Linear Algebra I

Suggestions for Review: Final Exam

The following is more of a review sheet for the material after Midterm II than it is one for the final exam. Iwould suggest using the review sheets for the two midterm exams along with this one, as review sheets forthe final exam.

I suggest before you go through this document that you read through all your class notes and/or the textbookand, as youre reading, solve questions as you encounter a new topic. You should at least attempt to solveall the computational style questions from the textbook, then try some of the Find an Example questionsand some of the True or False questions. Once youve systematically worked through all the material fromthe beginning, only then read this document. Organizing a carrying out your own review is a critical stepto understanding and mastering new material. It is much more valuable than going through someone elsesreview. You should be able to state theorems and definitions precisely and make connections between variousideas. You should be able to solve particular examples explaining you reasoning at every step. If you aredoing a proof and you are using a theorem to help complete the proof, you should state the theorem precisely.Writing something like By Theorem in the textbook, yadda, yadda, yadda... is not acceptable. Workingin small groups may worthwhile.

This may not cover every idea you need to master (not that Im intentionally omitting something) but it isintended to identify and solidify the main points and connections between various concepts.

I hope this helps. Good Luck!

Chapter 8: Orthogonality

Keywords: Dot product, norm (length) of a vector, orthogonal, orthogonal vectors, orthogonal complement,orthogonal set, orthogonal basis, orthogonal projection onto a vector, orthogonal projection onto a subspace,Gram-Schmidt process, orthonormal set, orthonormal basis, normalizing.

Define what it means for the set {v1, . . . ,vk} of vectors in Rn to be orthogonal. Define what it meansfor the set to be an orthogonal basis for a subspace W of Rn. What is the difference between anorthogonal basis and an orthonormal basis?

Does every subspace of Rn have an orthonormal basis? What about the zero subspace? Given a basisfor a subspace W of Rn, how do you find an orthonormal basis? Describe the Gram-Schmidt procedure.

What are the essential properties of a projection onto a subspace W of Rn. How do you calculateprojW (x) for any x Rn? What is projW (x) if x W? What is projW (x) if x W?

Practice Problems

1(a) Consider the subspace W ={x1x2

x3

|x1x2x3

101

= 0} of R3. Express the vector x =21

3

as thesum of a vector in W and a vector in W.

(b) Let W be a subspace of Rn and let x Rn. Prove that x = projW (x) + projW(x).

1 of 4

2. Let =

{2111

,

1334

,

1101

,

1231

}

.

(a) Show that is an orthogonal basis for R4.

(b) Normalize the vectors in to produce an orthonormal basis for R4.

(c) Let x =

1213

. Find [x] - the coordinate vector of x with respect to - and show that ||x|| = ||[x] ||.(d) Prove a more general version fo part (c): Let {v1, . . . ,vk} be an orthonormal set of vectors in Rn and

let x = x1v1 + + xkvk. Prove that ||x|| =x21 + . . . x

2k.

3(a) Let x1 =

1111

, x2 =

1210

, and x3 =

1113

. Show that {x1,x2,x3} is an orthogonal subset of R4 andfind a fourth vector x4 such that {x1,x2,x3,x4} is an orthogonal basis for R4. To what extent is x4

unique? (Hint: Let x4 =

xyzw

and solve the system of equations defined by xi x4 = 0 for 1 i 3.)3(b) Prove a general version of 3(a): Let {x1,x2,x3} be an orthogonal subset of R4. Prove that you can

always find a fourth vector x4 such that {x1,x2,x3,x4} is an orthogonal basis for R4. To what extentis x4 unique? Prove your answer. (Hint: What is the rank of the coefficient matrix for the system oflinear equations defined by xi x4 = 0 for 1 i 3?)

4. Let S1 =

{1211

,

0111

}

, and S2 =

{1122

,

31344

}

. Note that 1 and 2 are orthogonal subsets

of R4.

(a) Show that S1 and S2 both span the same subspace W of R4. In other words show that W = Span(S1)iff W = Span(S2).

(b) Taking W = Span(S1), find projW (

1111

).

(c) Taking W = Span(S2), find projW (

1111

). You should get the same answer as in part (b). Why?

2 of 4

(d) Let x =

x1x2x3x4

. Taking W = Span(S1), find projW (x) and a matrix A such that projW (x) = Ax.

(e) Let x =

x1x2x3x4

. Taking W = Span(S2), find projW (x) and a matrix A such that projW (x) = Ax. Youshould get the same matrix A as in part (d). Why?

(f) Show that the matrix A from part (d) satisfies A2 = A. Explain this result geometrically.

5. Let x Rn and let p be the orthogonal projection of x onto W where W is a subspace of Rn. Provethat for all y W ,

||x (p+ y)||2 = ||x p||2 + ||y||2.It follows that p is the closest point in W to y. It may help to draw a picture and interpret the resultgeometrically.

Chapter 7: Vector Spaces

Keywords: Vector space, additive inverse, additive identity (zero vector), subspace, trivial subspace, linearcombination, span, linear independence, linear dependence, basis, standard basis, dimension.

State the essential properties of a vector space. Why is the empty set not a vector space? What is a basis? What are some of the important properties of a basis? What is the dimension of a vector space V ? Explain how knowing the dimension of a vector space is

helpful when you have to find a basis for the subspace.

State the formal definition of linear independence and linear dependence. Why is linear independenceimportant when looking at spanning sets?

Explain how you use information about consistency of systems of equations and uniqueness of solutionsin determining if a vector belongs to a given subspace and in testing if a set is linearly independent.

Note how the concepts of span, linear independence, subspace, basis, and dimension extend fromEuclidean space (Rn) to a general vector space V .

Practice Problems

6. Let V be the set of ordered pairs (x, y) of real numbers with the operations of vector addition andscalar multiplication given by

(x, y) + (x, y) = (y + y, x+ x)c(x, y) = (cx, cy)

V is not a vector space. List one of the properties from the definition of vector space that fails to holdfor V . Justify your answer.

3 of 4

7. Let V = R22 be the vector space of all 2 2 matrices

V = {(a bc d

)| a, b, c, d R}.

For each of the subsets W of V listed below, determine if it is a subspace of V . If it is, show that allthree conditions in Denition 7.3 of the textbook are satisfied; if it is not, show by example that one ofthe conditions fails to hold.

(a) W = {(a bc d

)| a+ b = 0}.

(b) W = {(a bc d

)| ab = 0}.

8. Let V = P 4 be the vector space of all real valued polynomials of degree less than or equal to four. LetW = {p(x) P 4 | p(2) = p(2)}.

(a) Show that W is a subspace of V .

(b) Find a basis for W .

9. Find all values of c such that the set S = {x3 +x+ 1, x3x2 + 1, x3 + cx2 + cx} is linearly independentin P 3 (the vector space of all real valued polynomials of degree less than or equal to three.)

10. Let S be the subset of R22 (the vector space of all 2 2 matrices) given by

S = {[

2 12 1

],

[0 32 1

],

[c c+ 20 0

]}

Find all values of c for which S linearly dependent.

11. Consider the set S ={[1 1

0 1

],

[2 11 1

]}.

(a) Show that any matrix[a bc d

]with a b+ c 6= 0, or 2a 3b+ d 6= 0 is not in Span(S).

(b) Find a basis for R22 (the vector space of all 2 2 matrices) that contains the set S.

12. Suppose that W is a subspace of a vector space V , and let y V be a fixed vector. Show the setS = {x V | x = y +w for some w W} is a subspace of V if and only if y W .

4 of 4