Solution
= Solution
\b[No.] Take the one-by-one <matrices> $A=[1]$, $B=[-1]$. Both <column spaces> are $C=\mathbb R$, whereas $A+B=[0]$ has <column space> $\{0\}$. More generally, taking $B=-A$ cancels any nonzero <column space>. One always has $\operatorname{Col}(A+B)\subseteq C$, because each sum column lies in the <vector subspace> $C$, but equality can fail through cancellation.