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"??

Tonio
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

.

OR

OR

These imply that

, and that they are independent.

As such,

and

are independent.

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Tell me if my answer needs any clean-up or reordering.