Math 2270
Quiz 5
1. Find vectors that span the kernel of the matrix

1 2
A= 1 3
3 2

3
2 .
1
To do this, we can row reduce A in order to solve the homogeneous equation A~x = ~0:

 
 
 

1 2 3
1 2
3
1 0
5
1 0 0
 1 3 2  ∼  0 1 −1  ∼  0 1 −1  ∼  0 1 0  .
3 2 1
0 −4 −8
0 0 −12
0 0 1
That rref(A) = I implies the solution to A~x = ~0 is x1 = x2 = x3 = 0, and this solution is unique. Therefore,
ker(A) = {~0} = span(~0).
2. Consider a nonempty subset W of Rn that is closed under addition and under scalar multiplication. Is W
necessarily a subspace of Rn ? Explain.
It is. By assumption, W satisfies two of conditions for being a subspace. We only have to check that ~0 ∈ W .
Since W is nonempty, choose ~x ∈ W . Since W is closed under scalar multiplication, the vector k~x ∈ W for any
choice of scalar k ∈ R. In particular, if we choose k = 0, then ~0 = k~x ∈ W . This shows that W is a subspace and
that the requirement ~0 ∈ W is unnecessary as long as the other two requirements are satisfied.