# A question about direct sums of subspaces…

Let $W_1$ be a subspace of a finite dimensional vector space $V$. Prove that there exists a subspace $W_2\subset V$ such that $V=W_1\oplus W_2$.

## EDIT$^1$:

This may be of use here:

What does $$W_1 \cap W_2 = \{0\}$$ even mean? The zero vector is the only thing shared in common by the two subset?

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I don't see how this "could mean various things". Can you elaborate on where you think the ambiguity is? –  Alex Becker May 22 '13 at 5:47
$W_1$ could be the set $\{0\}$, and so could $W_2$, right? –  Trancot May 22 '13 at 5:49
In that case, if $V$ is not trivial, then $W_1 + W_2 \neq V$. –  Adam Saltz May 22 '13 at 5:51
I mean it could be, so it's hard to generalize, right? –  Trancot May 22 '13 at 5:52
I'm not sure what any of your questions are asking, to be honest. I think you should review the definitions of the terms and symbols in the question. BTW, you are correct that $W_1\cap W_2=\{0\}$ means that the only thing shared by the two subspaces is the zero vector. –  Alex Becker May 22 '13 at 6:02

Choose a basis for $W_1$. This basis can be extended to a basis for $V$ (well-known result). Now let $W_2$ be the span of the newly added vectors. Then $W_1 \cap W_2 = \{0\}$ by linear independence and $W_1 + W_2 = V$.

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Yes, Jesus, that is precisely what is stated in Cohn's work under the name complement, right? –  Trancot May 22 '13 at 6:59
Yeah, seems so. –  Jesus May 22 '13 at 7:01
Ah, you beat me to it. Deleted my answer. (+1) –  Bryan Urízar May 22 '13 at 7:03

Consider the set $\mathcal W$ of subspaces $W\subseteq V$ with $W_1\cap W=\{0\}$. Let $W_2$ be a maximal element of $\mathcal W$ (with respect to inclusion). Why does such maximal element exist and why does that solve the problem?

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Here are some explicit examples for you to play around with:

• Let $V$ be 2-dimensional, say $\langle x, y\rangle$, and let $W_1 = \langle x\rangle$. Can you see that $W_2 = \langle y\rangle$ works? Can you see that $\langle x+y\rangle$ also works, and that $\langle 3x - 5y \rangle$ also works, but that $\langle x \rangle$ doesn't, that $\{0\}$ doesn't, and that $\langle x+y, x-y\rangle$ doesn't? (Draw pictures.)
• Let $V = \langle x_1, \dots, x_n\rangle$ (n large), $W_1 = \langle x_1, x_2, x_3\rangle$, and show that $W_2 = \langle x_4, \dots, x_n\rangle$ works.
• Let $V = \langle x, y, z\rangle$, and let $W_1 = \langle x\rangle$. Can you pick the obvious space that works as $W_2$? Can you pick a non-obvious space that works? Can you show why $\langle y\rangle$ and $\langle 3x+5y, x-y\rangle$ don't work?
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