Math 220
Section 4.2 Practice
Names: _______________________________________
1. Use an appropriate Theorem to show that the set V is a vector space
or find a specific counterexample to show that is not a vector space.
 x 



V   y  : x  y  0; y  z  0
 z 

 

Hint: Since both equations are set equal to zero, try writing V as the null
space of a matrix A.
There are two ways to do this:
1. With the hint: Write the restrictions on x, y, and z as a system of linear
equations, then as the associated matrix equation Ax = 0.
x y
0
yz 0
 x
1  1 0   0
0 1 1  y   0

 z   
 
Since V is the set of all solutions to the homogeneous equation Ax = 0,
V = Nul A. Hence, by Th. 4.2, V is a subspace of R3, therefore a vector
space.
2. Without the hint: Since x – y = 0, x = y. Also, since y + z = 0,
z = -y = - x. Then any vector v in V can be written
 x
 1
v   x   x  1
  x 
  1
 1 


V  Span  1 
Thus,
  1  , so V is a vector space by Th. 4.1.
  
2. Let A  1 9  3 0 5 . Fill in the blanks with the appropriate
vector space: R1, R2 , R3 , …
Col A is a subspace of ___ R1______.
Nul A is a subspace of ___ R5______.
3. Find a non-zero vector in Nul A and non-zero vector in Col A. Use
the back of this sheet if you need extra room. You do not need to show
individual row operations.
9
 3
 1

3

A
 2 6 


 4  12
Nul A is the set of all solutions to the matrix equation Ax = 0, so we
solve the equation using the augmented matrix [A 0].
9 0 1  3 0
 3
1
 0

3
0
0
0
~

A
 2  6 0  0
0 0

 

0 0
 4  12 0 0
 x1  3x2
The solution is  x is free .
 2
3
v

1 .
To get a non-zero vector v in Nul A , let x2 = 1, then x1 = 3, and
 
Col A is the set of all linear combinations of the columns of A.
 if u  Col A, and c1, c2  R,
 3
 9
  1
 3



u  c1
 c2 
 2
 6 
 


4

12
 


Let c1 = 0, c2 = 1.
 9
 3
  Col A
u
.
 6 



12

