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 thedirect sum of and by , and denote theexternaldirect sum of and by . Define by for all .internal

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.