M340L Exam 2 Practice Problems
(1) True/False. Please mark a T or F beside each question. No explanation is necessary
(although in some cases a short calculation/drawing may be useful!)
(a) If A is any matrix then its row space equals its column space.
(b) If A and B are both n × n matrices then det(A + B) = det(A) + det(B).
(c) If A is a 3 × 3 matrix with det(A) = 0 then the columns of A are linearly dependent.
(d) If a 5 × 4 matrix A has rank r = 3 then its null space is one-dimensional.
(e) If two matrices are row equivalent, then their null spaces are the same.
(f) If a vector space V is 5 dimensional, then every collection of 4 vectors is linearly
independent.
(2) Suppose A is the matrix

1
0

A=
2
0
3
0
4
2
2
0
3
0

5
3

.
1
1
(a) Calculate det A.
(b) What is the rank of 5A?
(c) What is the rank of A3 ?
(d) Suppose the matrix B is obtained from A by switching row 1 and row 2. What is
det B?
1
2
 
x1
x 
 2
(3) Suppose H is the subset of R4 consisting of all vectors ~x =   satisfying
x3 
x4
x1 + x2 + x3 + 2x4 = 0,
x2 + x3 = 0.
(a) H is a subspace of R4 . Briefly describe how we knew that (HINT: think null spaces!).
(b) What is the dimension of H? Find a basis.
(4) Suppose A is the 4 × 5 matrix given

1
0

A=
2
0
below.
2
1
6
4

0 1
4
1 −1 1 


2 1 11
4 −7 1
(a) Find a basis for Col(A). What is the dimension of Col(A)?
(b) Find a basis for Row(A). What is the dimension of Row(A)?
(c) What is the dimension of N ul(A)?
3
(5) Suppose B = {~b1 , ~b2 , ~b3 } and C = {~c1 , ~c2 , ~c3 } are both bases for a 3 dimensional vector
space V . Suppose ~c1 = 2~b1 − ~b2 + ~b3 , ~c2 = 3~b2 + ~b3 , ~c3 = 4~b1 + 2~b3 .
(a) Find the change-of-coordinates matrix from C to B.
 
1
 
(b) If [~x]C = 2, find [~x]B .
3
(6) Short answer.
(a) Suppose V is a vector space with dim V = 3, and suppose {~v1 , ~v2 } are vectors in V .
What are the possibilities for the dimension of Span{~v1 , ~v2 , ~v1 + ~v2 }?
"
(b) Consider the subset H of
R2
#
x1
given by all vectors ~x =
satisfying x1 ≥ 0. Is H
x2
a subspace of R2 ?
#
0.5 0.25
(c) Find a nonzero steady state vector for the stochastic matrix A =
.
0.5 0.75
"
(d) Use determinants to find the area of the parallelogram with vertices (−1, 0), (1, 1),
(−1, 2), (1, 3).
(e) Recall that P3 is the vector space of all polynomials p(t) with real coefficients of
degree at most 3. Define a linear transformation T : P3 → R3 by


p(0)


T (p) = p(1)
0
. Find a basis for the kernel of T .