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If $A$ and $B$ are linear maps of finite rank from $V$ to $W$, where $V$ and $W$ are vector spaces, how can you find out something like rank$(A + B)$ or rank$(A - B)$ simply from that?

I suppose $(A - B)(v) = A(v) - B(v)$, but then can you determine the image of $A - B$ (and thus the rank) simply from that?

Considering a matrix interpretation didn't help either as if the matrices of $A$ and $B$ are of different dimensions, can you even subtract them?

ENPM
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1 Answers1

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The rank of $A+B$ can take any possible value, that can be related to the rank of $A$ and $B$. Imagine

$$A=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ in a certain basis.

  • If $B=-A$ then the rank of $A+B$ is zero.
  • If $B=\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ then the rank of $A+B$ is one.
  • If $B=\begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}$ then the rank of $A+B$ is two.

(Note that $A$ and $B$ have rank two.)

mfl
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  • Ok, so is there any way to know rank$(A - B)$ for a general $A$ and $B$ (with respect to $A$ and $B$ obviously)? What happens if the two functions have different rank in the first place? – ENPM Oct 09 '16 at 18:09
  • You will have a wide rank of possibilities again. But you have some bounds: rank$(A+B)\le$rank$A+$rank$B.$ See http://math.stackexchange.com/questions/851596/show-operatornameranka-operatornamerankb-ge-operatornameranka – mfl Oct 09 '16 at 18:15
  • Ok, I think I have what I need now. Thank you! – ENPM Oct 09 '16 at 18:20