What does the $\operatorname{rank}(f(T))$ mean exactly, if $f:L(V,W) \to L(V,W)$ and $T \in L(V,W)$, where $L(V,W)$ is the set of all linear transformations between two finite dimensional vector spaces $V$ and $W$?
What is the rank of a linear transformation space? $L(V,W)$
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linear-algebra
vector-spaces
linear-transformations
matrix-rank
1 Answers
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Well, $f$ is a map assigning to each linear map $T$ a linear map $f(T)$. Therefore, $\operatorname{rank}(f(T))$ is the rank of said linear map $f(T)$: $$\operatorname{rank}(f(T))=\dim\left( f(T)(V)\right)$$
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0Are you sure its not the dim R($f(T)$)? Is that just all $T$ it encompasses which would be $L(V,W)$ – 2017-02-12
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0In my opinion, if $f$ is a linear map, the rank of $f$ (id est, $\dim (f(L(V,W)))$ or $\dim\operatorname{im} f$) would just be $\operatorname{rank} f$. @BBest – 2017-02-12