I'm having a silly doubt studying linear algebra, can someone help me out? Let me first state it formally.
Let $V$ be a finite-dimensional $\mathbb{K}$-vector space. Let $W,V_1,\ldots V_k$ be subspaces of $V$ such that $$ V= V_1\oplus V_2\oplus \ldots \oplus V_k.$$
I want to know if the statement
$$ W = (W\cap V_1)\oplus (W\cap V_2)\oplus\ldots\oplus(W\cap V_k).$$
is true!
The inclusion "$\supseteq$" is trivial. On the other hand, I don't know how to show "$\subseteq$". I thought that a proof by induction would help me here but couldn't think of anything useful.
Does anybody even know if the result is true?
It is false.
Let $k=2,\ a\in V_1,\,b\in V_2$ nonzero vectors and $W:=\mathrm{span}(a+b)$.
Then $W\cap V_i=\{0\}$, else $b\in V_1$ [resp. $a\in V_2$] would follow.