I have a good deal of work done for this, but could use a second opinion (and could use some info on whether I'm missing anything).
Suppose that and are vector spaces over a field .
Let and be subspaces of a finite-dimensional vector space .
Prove that and are independent if and only if .
Here is my work so far (well, it's how my Professor showed us during his office hours, so he probably made it incomplete).
and such that , then
If we suppose that are independent, this implies that .
Suppose that ,
Clearly, (We can let and )
Using the above, we now need to prove that are independent.
Suppose and and that .
These imply that , and that they are independent.
As such, and are independent.
Tell me if my answer needs any clean-up or reordering.