Proofs where V is a finite-dimensional vector space

These are probably easier than I think they are.

Suppose and are vector spaces over a field .

Now suppose are subspaces of and that is finite-dimensional.

**a)** Prove that .

The question recommends proving this via induction (although proving via bases is possible, albeit more difficult). I think what I have below works, but if it needs correction, I'd like to know.

For is true. (base case)

Assume the result is true for ,

i.e. (hypothesis)

For , (inductive case)

Therefore,

**b)** Prove that if and only if every vector is equal to for a *unique* choice of .

I don't have an answer yet for this half, but I'm working on it.

Any help on these would be appreciated.