Is any subspace of a direct sum necessarily a direct sum of subspaces?

Question

If I have a direct sum $V = V_1 \oplus V_2$ and a subspace $W \subset V$, it it necessarily true that $W = W_1 \oplus W_2$ where $W_1 \subset V_1$ and $W_2 \subset V_2$?

I believe this is true since we should be able take $W_1 :=W \cap V_1$ and $W_2 := W \cap V_2$, but I just want to make sure there isn't a flaw this this argument. Thanks!

Answer 1

In your attempt, while the sum $W_1 + W_2$ is direct (that is, $W_1 \cap W_2 = 0$), it might not equal all of $W$.

There is a simple counterexample for $V = \mathbb{R}^2$. $V$ is the direct sum of the $x$ and $y$ axes, but if we set $W$ equal to the line $y = x$, then $W$ intersected with the $x$ and $y$-axes is zero.

Answer 2

No. Consider the simple case $$V_1, V_2 := \Bbb F, \qquad W := \langle (1, 1) \rangle \subset V := \Bbb F \oplus \Bbb F .$$ In this case, $W \cap V_a = \{ 0 \}$ for $a = 1, 2$, so your construction already fails for this example.

In fact, the only subspaces of $V_a$, $a = 1, 2$, are $0$ and $V_a$ itself, so the only $1$-dimensional subspaces of $V$ that are the direct sum of a subspace of $V_1$ and a subspace of $V_2$ are $V_1$ and $V_2$ themselves, and neither of these is $W$. So, in general, there are not $W_a \subseteq V_a$, $a = 1, 2$, such that a general subspace $W \subset V_1 \oplus V_2$ is equal to $W_1 \oplus W_2$.

Answer 3

Note that $\langle(1,1)\rangle\subset\mathbb{R}^{2}=\langle (0,1)\rangle\oplus\langle (1,0)\rangle$ can not be written as a direct sum of subspaces of $\langle(0,1)\rangle$ and $\langle(1,0)\rangle$.