MATH 323.502
Examination 2
April 8, 2015
NAME
SIGNATURE
“An Aggie does not lie, cheat, or steal or tolerate those who do.”
This exam consists of 5 problems, numbered 1–5. For partial credit you must present your
work clearly and understandably and justify your answers.
The use of calculators is not permitted on this exam.
The point value for each question is shown next to each question.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 5 PROBLEMS ON
5 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1
2
3
4
5
Total
Points
Possible Credit
15
8
20
18
12
73
NAME
1.
MATH 323
Examination 2
Page 2
[15 points] For each statement below, write down whether it is true or false.
(a) The function L : R2 → R1 defined by L xy = (x−y)2 is a linear transformation.
(b) For a 2 × 3 matrix A, it is possible for the column space to have dimension 3.
(c) For a 2 × 3 matrix A, the dimension of N (A) can be 1, 2 or 3, but not 0.
(d) For a 3 × 5 matrix C, if for every b ∈ R3 the system Cx = b is consistent, then
the rank of C is 3.
(e) Suppose that v1 , v2 , v3 are linearly independent vectors in R6 . Suppose that w1 ,
w2 , w3 are linearly independent vectors in R6 . Then together v1 , v2 , v3 , w1 , w2 , w3
are linearly independent vectors in R6 .
2.
[8 points] Let L : R2 → R2 be the linear
x1
3
L
=
x2
2
transformation defined by
1 x1 x2
1
.
4 x2 −x1 −1
Find a matrix A so that L(x) = Ax for all x ∈ R2 .
April 8, 2015
NAME
3.
MATH 323
[20 points; (a) & (b) 6 pts. each; (c) & (d)

1

3
A=
−1
Examination 2
Page 3
4 pts. each] Let A be the matrix

0 1
6 0 ,
−6 2
let V be the column space of A, and let W be the row space of A.
(a) Find a basis for V . Show your work.
(b) Find a basis for W . Show your work.
(c) What is the rank of A? Explain.
(d) What is the dimension of the null space of A? Explain.
April 8, 2015
NAME
4.
MATH 323
Examination 2
Page 4
[18 points] As usual we let S = {e1 , e2 } denote the standard basis for R2 . Suppose
that B = {u1 , u2 } is another basis for R2 , and suppose we have a change of basis
matrix
4 7
B
P = [I]S =
.
1 2
That is, for every x ∈ R2 , we have [x]B = P [x]S .
(a) Find a matrix Q such that for every x ∈ R2 , [x]S = Q [x]B .
(b) What are u1 and u2 with respect to the standard coordinates on R2 ?
(c) Suppose that L : R2 → R2 is a linear transformation and that with respect to the
basis S we have [L]SS = ( 01 30 ). Find [L]BB .
April 8, 2015
NAME
5.
MATH 323
Examination 2
Page 5
[12 points] Let S1 and S2 be finite subsets of a vector space V such that S1 ⊆ S2 . Let
d1 be the dimension of Span(S1 ), and let d2 be the dimension of Span(S2 ).
(a) Prove that d1 ≤ d2 .
(b) Prove that if Span(S1 ) = V , then d1 = d2 .
April 8, 2015