My information for this one is limited, so I'll need a hand.

Suppose $\displaystyle V$ and $\displaystyle W$ are vector spaces over a field $\displaystyle F$.

Now suppose $\displaystyle W_1$ and $\displaystyle W_2$ are subspaces of $\displaystyle V$. We denote thedirect sum of $\displaystyle W_1$ and $\displaystyle W_2$ by $\displaystyle W_1\times W_2$, and denote theexternaldirect sum of $\displaystyle W_1$ and $\displaystyle W_2$ by $\displaystyle W_1\oplus W_2$. Define $\displaystyle \varphi : W_1\times W_2\rightarrow W_1+W_2$ by $\displaystyle \varphi(w_1,w_2)=w_1+w_2$ for all $\displaystyle (w_1,w_2)\in W_1\times W_2$.internal

a)Prove that $\displaystyle \varphi$ is a surjective linear transformation.

b)Prove that $\displaystyle \varphi$ is an isomorphism if and only if $\displaystyle W_1+W_2=W_1\oplus W_2$.

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.

$\displaystyle V=W_1\oplus ...\oplus W_k$, if $\displaystyle W_1+...+W_k=V$ and $\displaystyle W_1,...,W_k$ are independent.