Dr. Byrne
Fall 2009
Math 237
Section 1
Linear Algebra Practice Exam #2
Covering Chapter 2
2.1 The Test for Linear Independence
2.2 Dimension
2.3 Row Space and Rank Nullity Theorem
And Chapter 3:
3.1 Linearity Properties (Transformations)
3.2 Matrix Multiplication (Compositions)
3.3 Inverses
3.5 The Matrix of a Linear Transformation
This practice exam is twice as long as the exam will be. The exam will have 5 ‘short’ questions
and 5 ‘long’ questions.
1. Which of the following vectors is dependent upon 2 4 ?
a. 0 0
c. 0 1
b. 1 0
d. 1 1
2. Which of the following vectors is in the span of 2 4 ?
a. 4 4
c. 0 4
b. 4 0
d. 4 8
3. Any basis for the vector space of 2x3 matrices M(2,3) will contain how many vectors? _____
4. Write down a basis for the vector space 6.
0 1 0 2 
5. Let A  0 0 1 0 . What is the rank of A? ______
0 0 0 1
1 2 3 4 
6. Let A  5 6 7 8  .
6 8 10 12
The rank of A is 2. What is the dimension of the column
space and what is the dimension of the nullspace?
7. In a linear system, there are fewer equations than unknowns. How many solutions may the
system have?
8. In a linear system, there are more equations than unknowns. How many solutions may the
system have?
9. In a linear system Ax=B, the nullity of A is 3, how many solutions may the system have?
 2 2
1
10. Let A = 
. What is the image of x=   under the transformation T: x  Ax?

1
  2 2
11. Use the test for linear independence to determine if the set S below is linearly independent.
1 1 2 2 1 1 1 0 1 
S  
, 1 0 1, 1 1 2 
2
1
1




12. Show that the set S below forms a basis for M(2,2).
1 0 1 1 1 1 1 1 
S  
,
,
,




0
0
0
0
1
0
1
1





13. Show the set S={1+x,x+x2,1+x2} forms a basis for the vector space of polynomials 2 =
{a+bx+cx2 | a,b,c  }.
14. Find a basis for the span of the set S below.
  2   0  0   4  
        
 2 1 0 4 
S   ,  ,  ,   
3 2 1 6 
3 1 3 6 
 x   4 xy
15. Show that the transformation T: 2  2 defined by T     
 is not linear.
 y    y 
 x  
   x
16. Show that the transformation T:    defined by T  y      is linear.
 z    z 
  
3
2
1 0 0 
17. Find the inverse of A  2 1 0 .


3 2 1
18. Multiply the matrices A and B to find the product A*B:
0 1
1 0 0 


A  2 1 0 , B  2 3
4 5
3 2 1
 x1  2 x2  x3 
 x1  
2 x1  4 x3 


3
4

19. Define T:   by T ( x2 ) 
    x  x  2x  .
2
3
 x3   1

3x1  2 x2  5 x3 
Find a matrix A such that T is
left multiplication by A.
20. For each part below, consider transformations T: 2  2 (i.e., functions that map a vector
from 2 to 2.)
a) Find a matrix A such that left multiplication by A dilates a vector by a factor of 2 in all
directions.
b) Find a matrix B such that left multiplication by B projects a vector onto the x-axis.
c) Find a matrix C such that left multiplication by C dilates a vector by a factor of 2 in all
directions and then projects it onto the x-axis.