.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 .
What is for you "independent linear subspaces"??
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.