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Unformatted text preview: . Sections L010], L020], L0202 and L5101 Last Name:
Given Name: Student Number: Faculty of Arts and Science
University of Toronto Final Examination, December 2006 MAT223H1F
Linear Algebra I A. Esterov, M. Shub, S. Uppal Duration: 3 hours No calculators or other aids are allowed. FOR MARKER USE ONLY 1 _/10
7 10f 14 Indicate your answer for the true/ false questions by shading in the box corre
sponding to your choice. Each correct answer is worth 2 marks, an incorrect answer is worth 1 mark,
and no answer is worth 0 marks.. [10] 1. Determine if the following statements are guaranteed to be true. (a) If A is a square, diagonalizable matrix whose characteristic polynomial is cA(/\) =
A3()\ — 3)2()\ — 4)3, the the nullspace of A might have dimension 2. C (b) If A = [a then A is diagonalizable if (a + d)2 — 4det(A) > 0. (c) The set of all n X n matrices A with det(A) = 0 is a subspace of the set of all n X n matrices. ((1) Let Y be a ﬁxed vector in R3. Then the linear transformation T: R3 »—> R3 deﬁned by T(X) = Y x X for every X e R3 is oneto—one . (e) If T: R" I—> Rm is a linear transformation and U is a subspace of R", then T(U) is a
subspace of R“. ' false 20f 14 Indicate your answer for the true/ false questions by shading in the box corre
sponding to your choice. Each correct answer is worth 2 marks, an incorrect answer is worth 1 mark,
and no answer is worth 0 marks.. [10] 2. Determine if the following statements are guaranteed to be true. (a) If A is an m X, n matrix with m < n, then the columns of A are linearly independent. (b) Suppose A is a 3 x 3 matrix Whose nullspace is a line through the origin in R3, then
the row space of A or the column space of A is also a line through the origin. (c) If V is a subspace of R” and W is a subspace of V, then Wi is a subspace of Vi. (d) If T: R3 I——> R3 is any linear transformation then ke'r(T) 76 im(T). (e) If T: R” H Rm is any linear transformation such that W” = span{T(X1),    , T(Xk)} then R" = span{X1,   ,Xk}. 30f 14 0 0 2
3.LetA= 0 O 0.
2 0 0 [5] (a) Find the eigenvalues and corresponding eigenspaces of A.
[3] (b) Find an invertible matrix P and a diagonal matrix D such that A = PDP‘I.
[4] (c) Compute A2233 where B = [1,1,1]T. 40f 14 EXTRA PAGE FOR QUESTION 3  Please do not remove 50f l4 [10] 4. Let U = {(x, 3/, 2,112) e R4  .7: + 2y + 32 + 4w = 0}. Find the orthogonal projection of
' (1,1,1,1) onto U. 60f 14 EXTRA PAGE FOR QUESTION 4  Please do not remove 70f 14 5. The two parts of this question are not related. [6] (a) Find the standard matrix of the linear transformation T: R3 H R3 given ker(T) =
span{[1, 1, IF, [1, —2, IF} and T([3,2,1]T) = [10,10,10]T. [6] (b) Let T: R2 H R4 be a linear transformation induced by the matrix A = . Find whme
H1000“: a vector X E R2 such that T(X) is as close as possible to [4,6, 6, 4]T. 80f 14 EXTRA PAGE FOR QUESTION 5  Please do not remove 90f 14 6. Let W = E P3(R)  p(—3) = 0}.
[5] (a) Show that W is a subspace of P3(R).
[5] (b) Find a basis for W and determine dim(W). Justify your answer. 10 of 14 EXTRA PAGE FOR QUESTION 6 _Please do not remove. 11 of 14 [8] 7. Let A be a 3 x 3 invertible matrix with det(A) = 10 and cofactors 012(A) = 5, 022(A) = 4, and 032(A) = {131 2
A $2 = 5
$3 ﬁnd :32. Justify your answer. 12 of 14 [8] 8. Suppose A is an m x n matrix and dim(null(A)) = d. Determine dim(null(AT)).
Justify your answer. 13 of 14 ll EXTRA PAGE FOR ROUGH WORK  Please do not remove 14 of 14 ...
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This note was uploaded on 01/19/2010 for the course MAT MAT223 taught by Professor Uppal during the Spring '09 term at University of Toronto Toronto.
 Spring '09
 UPPAL
 Linear Algebra, Algebra

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