Question

Show that rank(A+B) is never greater than the sum of rank(A) and rank(B).

Answer 1

Question 23816 Show that $\operatorname{rank}(A + B)$ is never greater than the sum of $\operatorname{rank}(A)$ and $\operatorname{rank}(B)$.

Answer. Assume that $A, B$ are matrix from $V = R^n$ to $W = R^m$. Next, by the definition $\operatorname{rank}(A + B) = \dim((A + B)V) = \dim(AV + BV) \leq \dim(AV) + \dim(BV) = \operatorname{rank}(A) + \operatorname{rank}(B)$, since dimension of sum of vector spaces is less than sum of dimensions of the respective terms.