Consider the vector space consisting of all linear transformations on a vector space $V$, and let $A$ be the (left) multiplication transformation that sends each transformation $X$ on $V$ onto $PX$, where $P$ is some prescribed transformation on $V$. If $P$ has rank $m$, what is the rank of $A$?
$Source$: Halmos Finite demensional vector space, Section 50 Rank and nullity, exercise #3
Now from the question, I think that means $AX=PX$. If $P$ has rank $m$, then $A$ must have the same rank, $m$.
Am I right? I am quite sure I have miss something. I don't think Halmos would ask such a simple question.
Please can you help, thank you very much.
You're misunderstanding the question: we start with a vector space $V$. Then, we consider the vector space of all linear transformations from $V$ to $V$. To be clear, let's give this its own name, let's say $W$. (This space is normally denoted $\mathrm{End}(V)$, because it consists of endomorphisms of $V$.)
Again, just to be clear, each element of $W$ is a different linear transformation from $V$ to $V$.
Now, having chosen some specific $P:V\to V$, we consider a linear transformation $A:W\to W$. The rule defining $A$ is that when we input a linear transformation $X:V\to V$, we output the new linear transformation $PX:V\to V$, that is, the composition of the maps $$V\xrightarrow{\;\;X\;\;}V\xrightarrow{\;\;P\;\;}V\\ \xrightarrow[PX]{\qquad\qquad\qquad}$$ The question is asking you what the rank of the linear transformation $A:W\to W$ is. To start with, you should understand that $\dim(W)=\dim(V)^2$ (think about the dimension of a space of $n\times n$ matrices).