## Dimension of the Sum of Two Subspaces

## Problem 440

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.

Then prove that

\[\dim(U+V) \leq \dim(U)+\dim(V).\]

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.

Then prove that

\[\dim(U+V) \leq \dim(U)+\dim(V).\]

Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$.

Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that

\[x=\sum_{i=1}^nv^i(x)v_i\]
for any vector $x\in V$.