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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?

user26857
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cele
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    See http://math.stackexchange.com/questions/386373/operatornameranka-b-%E2%89%A4-operatornameranka-operatornamerankb?lq=1, http://math.stackexchange.com/questions/246941/prove-that-operatornameranka-operatornamerankb-ge-operatornamera, http://math.stackexchange.com/questions/375982/rank-of-the-difference-of-matrices, if you don't understand any of these answers, please be specific about the things you don't understand. – user37238 Jul 01 '14 at 14:18

1 Answers1

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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 $$Rank(A+B) \le Rank(A) + Rank(B)$$

Joel
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  • Ive never seen the notation $ran$ before. What does it mean? Is there a different notation for it? – cele Jul 01 '14 at 14:41
  • $Ran(A)$ is the range of $A$: $$Ran(A) = { Ax : x \in V }$$ where $V$ is the vector space $A$ acts on. – Joel Jul 01 '14 at 14:43