Examples of dim(V+W)+dim(V∩W)=dim(V)+dim(W). I saw a few posts that ask to prove this identity, but I'm still trying to understand what it means intuitively. Can anyone show a few examples of it?
 A: Think about the $x-y$ and $y-z$ coordinate planes in space. Each has dimension $2$. Their intersection is the $1$ dimensional $x$ axis. Their sum is the full $3$ dimensional space.
A: Let $V$ be a finite dimensional $k$-vector space and let $U_1,U_2 \subset V$ be subspaces. Then one can consider the vector space $U = U_1 + U_2$. Now a natural question is which dimension $U$ has and how that dimension depends on the dimensions of $U_1$ and $U_2$. The  observation is, that in general $\text{dim}(U) \neq \text{dim}(U_1) + \text{dim}(U_2)$. Consider for example:

The dimension of a subspace is always bounded by the dimension of the parent space, such that we have $\text{dim}(U) \leq 3$ in the case of $V = \mathbb{R}^3$. Therefore the dimension of the two planes cannot add up. One has to take the intersection into account and substract the dimension of the intersection.
In the case that the intersection is trivial, we call $U$ the direct sum of $U_1$ and $U_2$, which basically means that $U$ has a decomposition into $U_1$ and $U_2$. An equivalent way of stating that is that each vector $u \in U$ has a unique representation as $u = u_1 + u_2$, where $u_i \in U_i$. In that way the notion of a direct sum can be generalized to arbitrary finite sums of vector spaces $U_1 + \dots + U_n$.
