Let $k$ and $n$ be positive integers and let $F$ be a field. For matrices $A,B \in M_{k\times n} (F)$, show that the rank of $A+B$ is no more than the sum of the ranks of $A$ and $B$
I believe this question is addressed here, but to be honest I don't quite understand the explanations given. Can someone possibly help with a more detailed explanation?
The rank of a matrix is the dimension of its range. One way to demonstrate this inequality would be to show that the range of $A+B$ is a subset of the direct sum of the ranges of $A$ and $B$.
In particular, let $y \in Ran(A+B)$. This means that there is a vector $x$ for which $$y = (A+B)x = Ax + Bx.$$ Since we can express $y$ as a sum of vectors in the range of $A$ and $B$, this means that $$y \in Ran(A) + Ran(B).$$ It has now been demonstrated that $$Ran(A+B) \subset Ran(A) + Ran(B).$$
Finally notice that $$dim(Ran(A) + Ran(B)) \le dim(Ran(A)) + dim(Ran(B)) = Rank(A) + Rank(B)$$ we may conclude that $$Rank(A+B) \le Rank(A) + Rank(B)$$