I'm just starting out with linear algebra and I got to admit is quite an interesting topic. I've reached the point of subspaces and I got to this interesting formula: Let $V_1$ and $V_2$ be finite subspaces of the vector field $V$. Then $$\dim(V_1+V_2)=\dim V_1+\dim V_2-\dim(V_1\cap V_2)$$ First thing is I can't seem to think of a proof of the euqality by myself. All I can think of is an example: Let $V=\Bbb R^2, V_1=[a, 0], V_2=[0, b]$. Basically $X$ and $Y$ axis. They intersection is the point $(0,0)$. $\dim (V_1+V_2)=2$. So we have $2=1+1-0$. Which is in fact correct. However if I try to apply this for $\Bbb R^3$, all examples I can think of don't satisfy the equation. And I believe that it is because I take the intersection of two three-dimensional objects to be either another 3d object or a 2d plane.
Is that correct at all?
If not, what is the intersection of two three dimensional spaces?
How can I prove that $\dim (V_1+V_2)= \dim V_1+\dim V_2- \dim(V_1 \cap V_2)$