Tagged: sum of subspaces

Problem 440

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.
Then prove that
$\dim(U+V) \leq \dim(U)+\dim(V).$

Problem 430

Let $V$ be a vector space over a field $K$.
If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset
$W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}$ is a subspace of the vector space $V$.