MATH 323.501
Examination 2
November 7, 2013
NAME
SIGNATURE
“An Aggie does not lie, cheat, or steal or tolerate those who do.”
This exam consists of 6 problems, numbered 1–6. 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 6 PROBLEMS ON
6 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
1
2
3
4
5
6
Total
Points
Possible Credit
15
8
20
24
12
16
95
NAME
1.
MATH 323
Examination 2
Page 2
[15 points] For each statement below, write down whether it is true or false.
(a) The rank of a matrix B plus the dimension of the null space of B is equal to the
number of rows of B.
(b) If L : V → W is a linear transformation, then the kernel of L is trivial if and only
if L is surjective.
(c) Suppose V1 and V2 are subspaces of R4 with dim(V1 ) = 2 and dim(V2 ) = 3. Then
dim(V1 ∩ V2 ) is either 1 or 2.
(d) Suppose that v1 , v2 , v3 , v4 are linearly dependent vectors in Rn . Then we have
Span(v1 , v2 , v3 , v4 ) = Span(v2 , v3 , v4 ).
(e) If λ is an eigenvalue of a matrix A, then A2 − λA is singular.
2.
[8 points] Let L : R2 → R2 be defined by
x1
x2 − x1
L
=
.
x2
2 − x2
Is L a linear transformation? Justify your answer.
November 7, 2013
NAME
3.
MATH 323
[20 points; (a) & (b) 8 pts. each, (c) 4 pts.]

1 1

A= 3 5
0 1
Examination 2
Page 3
Let A be the matrix

−2 0
4 2 ,
5 1
and let V be the column space of A.
(a) Are the columns of V linearly independent or linearly dependent? Justify your
answer.
(b) Find a basis for V . Show your work.
(c) What is the rank of A? Explain.
November 7, 2013
NAME
4.
MATH 323
Examination 2
Page 4
1
[24 points] Let A = {( 11 ) , ( −1
)} ⊆ R2 . We can take it as given that A is a basis
2
for R . Suppose that B is another basis of R2 with change of basis matrix
2 5
P =
.
1 3
That is, for every v ∈ R2 , we have [v]A = P [v]B .
(a) Find a matrix Q such that for every v ∈ R2 , [v]B = Q [v]A .
(b) Suppose that L : R2 → R2 is a linear transformation and that with respect to the
1 −2
B
basis A we have [L]A
A = −3 4 . Find [L]B .
(c) What are the vectors in B with respect to the standard coordinates on R2 ?
November 7, 2013
NAME
5.
MATH 323
Examination 2
Page 5
[12 points] Suppose V is a vector space and that {u, v, w} is a basis for V . Prove
that {u + 2v + 3w, 4v + 5w, 6w} is also a basis for V .
November 7, 2013
NAME
6.
MATH 323
Examination 2
Page 6
[16 points] Consider the matrix


1 0 −2
A =  2 0 −4 .
−1 0 2
(a) Find the eigenvalues of A.
(b) For each eigenvalue λ of A, find vectors that span the eigenspace corresponding
to λ.
November 7, 2013