Refer to this thread.
Refer to this thread.
Not in general. If , then ; they are not the same otherwise.
To show that is a subspace of , you have to prove three conditions:
(1)
(2) If , then
(3) If and , then .
For (1), .
For (2), if , then pick such that and . Then
And for (3), you just use preservation of scalar multiplication.