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 ,
For , (inductive case)
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.