Direct sum counterexample

Question

Let $V$ be a vector space over a field $\mathbb{F}$, and let $W_i\ (i=1,...,4)$ be four distinct nonzero subspaces of $V$. Suppose that $W_i \cap W_j =$ {$0$} for all $i\ne\ j$. Is it true that the sum $$W_1+W_2+W_3+W_4$$ is a direct sum? Give a proof if $'yes'$ or a concrete counter example if $'no'$.

I know it is not a direct sum but I cannot provide a concrete example. Anyone has an idea?

Answer 1

Hint: Consider four distinct lines (passing through $(0,0)$) in $\mathbb R^2$.