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9‘3? NO AIDS ALLOWED. Family Name: THE FACULTY OF ARTS AND SCIENCE
University of Toronto MAT223H1Y
Linear Algebra I FINAL EXAMINATION, AUGUST 2007 Examiners: D. Brooke, J. Uren
Duration: 3 hours (Please Print) Given Name(s): (Please Print) Please sign here: Student ID Number: You may not use calculators, cell phones, MP3 players, or
PDAs during the exam. Please read through the entire
test before starting, and take note of how many points
each question is worth. Answers for True/ False questions
do not require justiﬁcation. For every other question, you
must completely justify your answers —— partial credit will
be given for partially correct work. Do not remove any pages from the exam booklet. m
/5
Problem 8= /9 Page 1 of 18 \ 10
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€19 Total: 100 marks [30]
[2] [2] [2] Final Examination: August 2007 MAT223H1Y Linear Algebra I 1. For each part, determine whether the given statement is ” TRUE” or ” FALSE”. Clearly
indicate your answer in the space provided. (a) Let U and W be subspaces of a vector space V. Then U U W is a subspace of V. (b) Suppose {u,v,w} is a basis for a vector space V. Then {u + v,u + 112,1) + w} is
also a basis for V. (c) If U is a subspace of R", then it is always the case that (U i)J' = U. (d) Let X = {X1,X2,...,Xm} be any linearly independent set in R”. Let Y =
{a1X1, a2X2, ..., ame} where (11, ..., am are nonzero scalars. Applying the Gram— Schmidt orthogonalization algorithm to either X or Y yields the same orthogonal
set {E1, E2, ..., (e) Let U be a subspace of R". Let {X1, ...,Xm} be an orthogonal basis for U and
let {Yb be an orthogonal basis for UL. Then {X1, ...,Xm,Y1,...,Yk} is an
orthogonal basis for R". Page 2 of 18 [2] [2] [2] Final Examination: August 2007 MAT223H1Y Linear Algebra I (f) Let a be a nonzero real number. Then T = [x + 0] deﬁnes a linear transfor— y y
mation T : R2 —> R2. (g) Suppose U and W are subspaces of a vector space V, dim(V) = 3 and dim(U) =
2 = dim(W). If U 7e W then dim(U n W): 1. (h) Let AX : B be a system of n linear equations in n variables. If AX = B has a
solution, then A has rank n. (i) Let A = . Then /\ = 3 is not an eigenvalue of A. 12
32 2 (j) LetA=[ 3 j! . Then X = [ ] is an eigenvector of A. Page 3 of 18 [2] [2] Final Examination: August 2007 MAT223H1Y Linear Algebra I (k) Rotation about any line through the origin in R3 is a linear transformation. (1) Let A be a 4 X 5 matrix. The columns of A are linearly dependent. (m) Let A and B be two n x n matrices. Then rank(A) S rank(AB). (11) Let U and W be subspaces of R”. If U Q W, then dim(WJ) S dim(UL). (0) Let A be an n X 71 matrix, and let u be any real number. Then det'(uA) = udet(A). Page 4 of 18 [15]
[3] [3] Final Examination: August 2007 MAT223H1Y Linear Algebra I 2. Give an example of each of the following. In each casei a brief explanation of Why your
answer is an example is all the justiﬁcation required: (a) A 3 x 5 matrix whose rank is 2. (b) A surjective (onto) linear transformation T : R4 —> R2. 1
(c) A nonzero subspace U of R3 such that 2 E U J'.
3 Page 5 of 18 [3] Final Examination: August 2007 MAT223H1Y Linear Algebra I (d) A matrix A whose characteristic polynomial is cA = m3 — x2 — 213. (e) A basis for M212 consisting only of matrices with the property that A2 = A. Page 6 of 18 [12] Final Examination: August 2007 MAT223H1Y Linear Algebra I —203 3. Let A = —3 1 3 . Find the characteristic polynomial, eigenvalues, and basic eigen
—4 0 5 vectors of A. Use this information to calculate the (2, 2)—entry of A726. Page 7 of 18 Final Examination: August 2007 MAT223H1Y Linear Algebra I Extra space for problem 3. Do not tear out this page. Page 8 of 18 [12] Final Examination: August 2007 4. Let A = pansion of 1
0
2
0
A. owHo 1
0
0
O Hoot—“O MAT223H1Y Linear Algebra I . Calculate the determinant of A without using a cofactor ex Page 9 of 18 Final Examination: August 2007 MAT223H1Y Linear Algebra I Extra space for problem 4. Do not tear out this page. Page 10 of 18 ...
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 Spring '09
 UPPAL
 Linear Algebra, Algebra

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