Complementary subspaces, True/False question

Question

True or False?

$W_1$, $W_2$ and $W_3$ are subspaces from the vectorspace $V$. If $W_1 ⊕ W_2 = V$ and $W_1 ⊕ W_3 = V$ , then $W_2 = W_3$.

I actually had this smaller question asked on a exam and I said it was true but I was later told it was false. Can somebody explain to me why so I can intuitively see it in my head that it's indeed false. Only then I can come up with a counterexample.

Thanks in advance.

Answer 1

$W_2$ and $W_3$ are isomorphic, but might not be the same subspace.

One way to look at this is to first choose a basis $B$ of $W_1$. There are different ways of extending this basis to a basis of $W_1 \oplus W_2$, so the additional vectors added to $B$ might span different subspaces.

Another way is to imagine an automorphism $\alpha$ of $V$, (i.e. $\alpha:V \to V$ is an invertible linear map). Suppose that $W_1$ is an invariant subspace of $\alpha$. Then $W_1 \oplus \alpha (W_2)=V$ for all such $\alpha$.

Answer 2

It's indeed wrong ! You have for example that $$\mathbb R^2=\text{Span}\{(1,0)\}\oplus \text{Span}\{(0,1)\}=\text{Span}\{(1,0)\}\oplus \text{Span}\{(1,1)\},$$ but $$\text{Span}\{(1,1)\}\neq \text{Span}\{(0,1)\}.$$