linear algebra section 4.2 subspaces
1. let w1 and w2 be subspaces of a vector space v . define w to be the set of vectors in v oftheformw1+w2 wherew1 ? w1 andw2 ? w2.
(a) show that w is a subspace of v .
(b) show that w1 and w2 are subspaces of w

Proof

Step-by-step explanation:

(a)

- We first have to show that . Since are subspaces, it holds that , hence .

- Second, we have to show that is closed under sum and scalar multiplcation. First, let us take .

Since a, b belong to , we can find and such that

Therefore,

since are vector subspaces of , it holds that

, which shows us that W is closed under sum.

On the other hand, let us take and . We already know that

where and . Moreover,

,

hence

belongs to , which shows that is closed under scalar multiplication.

(b) It is clear that . Moreover, since are subespaces of , they are closed under sum and scalar multplication, this propertie is remains true for as subsets of , which tells us that are subspaces of .