Our book states that there is an isomorphism from $Hom(V,W)$, the vector space of all linear transformations from $V$ to $W$, to the matrix space $M_{m\times n}^F$ which is defined by $T:\rightarrow [T]^B_C$ where $B$ and $C$ are bases of $V$ and $W$ respectively.
There are actually two questions here which I think are related.
It's not entirely clear to me what the nature of the object in $Hom(V,W)$ are. Are they the explicit formulas for $T$ according to the given basis? For example T(x,y)=(2x+y,x+y) according to the given basis?
This is related to another question I asked. Can the explicit formula for a linear transformation be according to anything other than the standard basis?