What is the meaning of pre-and post composing?

Question

In my linear algebra class my professor kept on using the words, precompose, and postcompose. Can someone verify if it is the case that:

Pre-composing $A$ with $B$ means $A \circ B$

and

Post-composing $A$ with $B$ means $B \circ A$?

Answer 1

No, exactly backwards, because the second operation goes on the left with functional notation. If you postcompose $A$ with $B$, that means you compose $A$ "after" or second. So $AB$. Precompose $A$ to get $BA$. $ABv$ means take a vector $v$, do operator $B$ to it, then afterwards do operation $A$ to the resulting vector.

Answer 2

I would say those wordings are ambiguous, because they include both of the maps, and there's nothing to make it clear which of them the "pre" or "post" speaks about. The fact that one of them takes a "with" doesn't really help.

My best advice would be to write "$A\circ B$" in symbols when there is any danger of misunderstanding. If you have to understand something someone else has written, you're unfortunately left to figure out what makes most sense in context.

A coarse rule of thumb would be that if $A$ is either something big and complex, or an arbitrary map you know nothing specific about, and $B$ is something simple which merely modifies the behavior of $A$, then "pre-compose" means to form $A\circ B$ and "post-compose" means $B\circ A$ -- no matter which is grammatically a direct object and which has a preposition.