Linear Algebra I · Vector spaces, subspaces, sums, and direct sums
The sum map detects an internal direct sum
Problem
Let $U,W$ be subspaces of a vector space $V$ over $K$, and define $S:U\times W\to V$ by $S(u,w)=u+w$. Prove $\ker S=\{(x,-x):x\in U\cap W\}$. Deduce that $S$ is injective exactly when $U\cap W=\{0\}$, surjective exactly when $U+W=V$, and bijective exactly when both conditions hold, equivalently when $V=U\oplus W$.
Hint
If $S(u,w)=0$, then $w=-u$; now use membership in both subspaces.
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