Prove that $V$ is a direct sum of subspaces

Question

Let $V$ be a finite-dimensional vector space and let $W_1,...,W_n$ be subspaces of $V$ such that $V=W_1+...+W_n$ and $\dim{V}=\dim{W_1}+...+\dim{W_n}$. Prove that $V=W_1\bigoplus...\bigoplus W_n$.

We have that $V$ is the direct sum of subspaces when the following hold:

1. $V=W_1+...+W_n$.
2. The subspaces are independent.

I'm trying to use the second condition that $\dim{V}=\dim{W_1}+...+\dim{W_n}$ to show that the subspaces are independent, but am having trouble with that.

Answer 1

Hint use the formula $ dim(V+W) =dim(V) +dim(W) - dim(V\cap W) $ and proceed recursively.

Answer 2

You have to prove that $W_i\cap W_j=\varnothing$ for every $i,j=1,\dots,n$ where $i\neq j$. (Iterate the Grassmann dimension formula $0=\dim(W_i\cap W_j)=\dim W_i+\dim W_j-\dim(W_i\cup W_j)$ then $W_i+W_j=W_i\oplus W_j$)