Q. (a):

, so your base for U is correct.

For V,W, the bases you provided are wrong; consider this: What is the image of ? (There is none, so these bases are wrong!)

The correct answers would be:

Can you see why these are the answers? I'll expand if you want me to.

Q. (b):

U+W is not direct as and

V+W is not direct as and

V+Uisa direct sum. Let us assume there exists a non-zero shared element between them, then:

so, assume , then , then but then and then

Q. (c): , ,

Yes, it is correct that if and A,B are subspaces of , then

Can you follow from here on which two spaces give the answer?

Spoiler: