Does a vector space equaling the direct sum of W and X as well as W and Y imply X=Y?

Question

Sorry if this is obvious, but if a vector space V equals the direct sum of subspaces W and X, and also equals the direct sum of subspaces W and Y, does that imply that X equals Y?

Answer 1

I assume you mean the internal direct sum: i.e. that $V = W+X$ and $W \cap X = 0$.

The answer is no; this is very rarely true. You can already see this in the plane $V = \mathbb{R}^2$; if $A$ and $B$ are any two distinct one-dimensional subspaces, then $V$ is the internal direct sum of $A$ and $B$, so taking any three distinct one-dimensional subspaces for $W,X,Y$ would give a counterexample.

You can prove $X$ and $Y$ are isomorphic, however; if $V$ is the internal direct sum of $A$ and $B$, then $A \cong V/B$ and $B \cong V/A$. So, given the premise, you have

$$ X \cong V/W \cong Y$$

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If you don't mean the internal direct sum (and you mean "isomorphic" rather htan "equal"), then you can't even prove $X$ and $Y$ are isomorphic. e.g. if $V$ is infinite dimensional, then $$ V \cong V \oplus V \qquad \text{and} \qquad V \cong V \oplus 0 $$

Answer 2

No. For instance take $V=\mathbb{R}^2$ and $W, X, Y$ any three different subspaces of dimension 1.