Decomposing vector space into direct sums

Question

I want to a find a vector space $V$ which has subspaces $W_1, W_2, W_3 \subset V$ so that $W_1 \neq W_2 \neq W_3$ and $$W_2 \oplus W_3 = W_1 \oplus W_3.$$

One idea I had is: \begin{align*} V & = \mathbb{R}^2 \\ W_1 & = \{(x,y) \in \mathbb{R}^2 \mid x \leq y \} \\ W_2 & = \{(x,y) \in \mathbb{R}^2 \mid x \leq y \} \\ W_3 & = \{(x,y) \in \mathbb{R}^2 \mid x = y\}. \end{align*} I am struggling to prove that this satisfies $W_2 \oplus W_3 = W_1 \oplus W_3$. Any help on how to approach this problem would be helpful.

Answer 1

If you allow $W_1 = W_2$, you can let $W_1$ and $W_3$ be any linear subspaces of a vector space whose intersection is $\{0\}$. If not, you can let $W_1$, $W_2$ and $W_3$ be any distinct one-dimensional subspaces of $\mathbb R^2$.

Answer 2

Take $W_1 = \text{span}\{ (1, 1) \}$, $W_2 = \text{span} \{ (1, 0) \}$, and $W_3 = \text{span} \{ (0, 1) \}$. Then $$W_1 \oplus W_3 = \mathbb{R}^2 = W_2 \oplus W_3$$