When and when isn't it permissible for matrix multiplication to a replacement for a linear transformation?

Question

I've been told that what a matrix of $T$ actually represents$^*$ can be a bit complicated, and that, when a mapping is from $F^m \to F^n$, the matrix of $T$, when multiplied by an arbitrary vector in the departure space, will give you the appropriately mapped vector in the arrival space. However, I've heard this isn't always the case, and that it isn't always the case that multiplying a vector by the matrix of $T$ will map it to the appropriate vector in the arrival space. I suspect that this is the case when $T : V \to W$ where $V$ and $W$ are not necessarily represented by $F^m$ and $F^n$ respectively. Can someone clear up this confusion for me? Can I always safely multiply a matrix of $T$ by a vector to find its image in the arrival space or not? And if not, why and when not?

$^*$Which is, to my mind, a representation of the mapping of a linear transformation $T: V \to W$ where $V$ and $W$ are not necessarily $F^m$ and $F^n$ respectively with a mapping of $F^m \to F^n$ since there is an isomorphism between $V$ and $F^m$ if $dim \ V = dim \ F^m$

Answer 1

In general, a linear mapping $T:V\to W$ is just defined as a function that satisfies certain conditions (i.e. linearity). There does not need to be any matrix involved in the definition of the map.

If $V$ and $W$ are finite-dimensional and you fix a basis of $V$ and a basis of $W$, you can always find a matrix that represents the map $T$ w.r.t. those bases. This always works, i.e. the matrix-multiplication is valid for all vectors of $V$. But:

• The matrix depends on the chosen bases. If you choose a different basis, the matrix changes even though the underlying linear map is the same. (For very simple vector spaces like $F^n$, there is a canonical standard-basis. For that reason the basis-dependance is often a bit overlooked)
• For infinit-dimensional vector-spaces, the story becomes much more complicated, and matrices are generally not used. (Might be possible to define an infinite matrix, but you have to talk about topology and convergence for that, which is outside the scope of basic linear algebra)