My information for this one is limited, so I'll need a hand.
Suppose and are vector spaces over a field .
Now suppose and are subspaces of . We denote the external direct sum of and by , and denote the internal direct sum of and by . Define by for all .
a) Prove that is a surjective linear transformation.
b) Prove that is an isomorphism if and only if .
There isn't anything in our reading material that differentiates between external and internal direct sums, though I found the definitions online. In case anyone needs the definition for direct sum (in a basic sense), I have it below.
, if and are independent.