 Solution

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 $\mathrm{Col}(A+B)\subseteq C$, because each sum column lies in the vector subspace $C$, but equality can fail through cancellation.