What is the meaning of pre-and post composing? 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 $B \circ A$?

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.