Exam 2B
Math 220
Name: ______________________________________
150
Please show all appropriate work clearly and write all explanations in
complete sentences.
Remember: Answers with no supporting work will receive no credit!
No calculators may be used on this exam.
1. Assume A = [a1 a2 a3 a4 a5] and B = [b1
equivalent, where
1
1
2 1
7
 1
0
  2 1  4

4

1

B
A
0
 3
1
6  5  1 and



1
2 2
5
0
 1
b2 b3 b4 b5] are row
0 2 0
1 0 0
0 0 1
0 0 0
0
9 
2

0
a. Find a basis for the column space of A. No explanations are
needed.
b. Find a basis for Nul A. No explanations are needed.
2. Is it possible that all solutions of a homogeneous system of 50
linear equations in 56 variables are multiples of one fixed non-zero
solution? Justify your answer. Mention any theorems that you use.
3. Let A be an n x n matrix. Write statements from the Invertible
Matrix Theorem that are each equivalent to the statement “A is
invertible”. Use the following concepts, one in each statement:
Col A
span
rank
4. Is the following set a subspace of R3? Justify your answer.
Mention any theorems that you use.
2a  4b  3c 


 : a, b, c  R 
W  
4


  9a  b  7c 





5. Let A be an m x n matrix. Prove Theorem 4.3: If A is an m x n
matrix, Col A is a subspace of Rm.
6. Mark each statement either true or false. You do not need to justify
your answer.
a) _____ A set of four vectors in P4 is a basis for P4.
b) _____ A linearly independent set in a subspace H of a vector
space V is a basis for H.
c) _____ The row space of A is the same as the column space of AT.
d) _____ A subset H of a vector space V is a subspace if and only if
H contains the zero vector.
e) _____ Row operations preserve linear dependence relations
among rows of A.
f) _____ The dimension of the null space of A is the number of
columns of A that are not pivot columns.
g) _____ If dimV = p, and p ≥ 2, then there exists a spanning set of
p + 1 vectors in V.
h) _____ A basis of a vector space V is a linearly independent set that
is as small as possible.
i) _____ Let   b1 , b 2 ,, b n  be a basis for the vector space V. Then
for each x in V, there exists a unique set of scalars c1 , c2 ,, cn such
that x  c1b1  c2b 2    cnb n .