Definition of direct sum

Question

I've a doubt on the definition of direct sum of two vector subspaces. Is it possible to state that

$$W_1\oplus W_2=V \iff \dim(V)=\dim(W_1)+\dim(W_2) \wedge W_1\cap W_2={ \vec{o} }$$

In particular my doubts are about the fact that the condition $W_1 + W_2=V$ is not mentioned. Is it implied in the two conditions in the statement?

Thanks in advice for your help

Answer 1

If $V \neq W_1 + W_2$, this means that there is a vector $v$ in $V$ which is not contained in the sum. But then it spans a subspace of dimension one, not contained in the sum $W_1 + W_2$, which contradicts the fact that $\dim V =\dim W_1 + \dim W_2$.