In my book of linear algebra, I am given the theorem:
Suppose V is a finite dimensional vector space and let W and U be the subspaces of V, then: dim(W + U) = dim(W) + dim(U) - dim(W $\bigcap$ U).
I didn't find a proof for this theorem in my book. How can I prove this, or how to start with it?
First, we give $W\times U$ a linear space structure: $$(w,u)+\alpha(w',u') :=(w+\alpha w',u+u').$$ This is called the direct product of two vector spaces $W\oplus U$. A simple problem is to contruct a basis of this space using bases of $W$ and $U$ proving that $\dim W\oplus U=\dim W+\dim U$.
Then we can consider a map:
$$L\colon W\times U\to V,~~L(w,u)=w-u.$$ This map is linear as: $$L((w,u)+\alpha(w',u'))=w+\alpha w'-u-\alpha u'=L(w,u)+\alpha L(w',u').$$
We quickly calculate its kernel and image:
Kernel: $L(w,u)=0$ iff $w=u$, that is $w\in W\cap U$, so $\ker L=W\cap U$.
Image: $\text{im } L =U+W$
Then using rank-nullity theorem: $$\dim W+\dim U=\dim W+U-\dim W\cap U.$$