# Working with external and internal direct sums

• October 6th 2010, 11:20 AM
Runty
Working with external and internal direct sums
My information for this one is limited, so I'll need a hand.

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

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

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

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

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.

$V=W_1\oplus ...\oplus W_k$, if $W_1+...+W_k=V$ and $W_1,...,W_k$ are independent.
• October 6th 2010, 01:18 PM
Bruno J.
$W_1+W_2 = \{w_1+w_2 : w_1 \in W_1, w_2 \in W_2\}$. It's a subspace of $V$ which contains both $W_1$ and $W_2$ (and in fact the smallest such subspace). In the event that every element of $W_2+W_2$ can be written uniquely as $w_1+w_2$, then we write $W_2\oplus W_2$ instead.

Both are direct consequences of the definitions. It's very easy to see that $\varphi$ is linear. To show that $\varphi$ is surjective, take any $w_1+w_2 \in W_1+W_2$; then $\varphi(w_1,w_2)=w_1+w_2$.

The second one isn't any harder! Give it a try and post your idea.