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MATH 3330–001
Matrices and Linear Algebra
Solutions to Quiz 6
Problem 1. (6 pts) Consider two subspaces V and W of Rn .
a. Is the intersection V ∩ W necessarily a subspace of Rn ? Justify your answer.
Solution: Yes, the intersection V ∩ W is a subspace of Rn . There are three things to
verify:
1. For any two ~x, ~y ∈ V ∩ W , must also ~x, ~y ∈ V and ~x, ~y ∈ W . Therefore ~x + ~y ∈ V
and ~x + ~y ∈ W which gives ~x + ~y ∈ V ∩ W .
2. For ~x ∈ V ∩ W and k ∈ R, must also ~x ∈ V and ~x ∈ W . Therefore k~x ∈ V and
k~x ∈ W which gives k~x ∈ V ∩ W .
b. Is the union V ∪ W necessarily a subspace of Rn ? Justify your answer.
Solution: No, the union V ∪W is not necessarily a subspace of Rn . For example n = 2,
and V consists of all vectors on x-axis and W consists of all vectors on y-axis. Then
1
0
1
6∈ V ∪ W.
∈ W, but ~x + ~y =
∈ V, ~y =
~x =
1
1
0

Problem 2. (4 pts) Find a

1 1

Solution: Let A = 1 1
2 2

1 1 1
basis of the image of  1 1 5 .
2 2 1

1
5 . The RREF of this matrix is
1


1 1 0
rref(A) =  0 0 1  .
0 0 0
Therefore a basis of A’s image is formed by
 
1
 1 ,
2


1
 5 .
1