Linear maps and matrix representations

Question

I have a question. I know that every linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ can be expressed as $T(x)=Ax$. Where A is a matrix. Does this hold for any basis or just the standard basis? I would really appreciate if someone tells me if it does or doesn't for a general basis and if it doesn't hold for nonstandard basis then what extra conditions are required for it to be represented in the form written above?

Answer 1

No, any basis will do.

These videos by Grant Sanderson may help you develop an intuition for why our choice in basis does not matter when describing a linear transformation.

To summarize, each element $v_i$ of a vector $\mathbf v$ in $\mathbb R^n$ essential describes "how many" of the $i$th basis vector to add up when constructing the vector $\mathbf v$. In $\mathbb R^3$,

$$ \begin{bmatrix} a \\ b \\ c\end{bmatrix} = ae_1 + be_2 + ce_3 $$

where $\{e_1,e_2,e_3\}$ is any basis of $\mathbb R^3$.

Each column of a matrix $A \in \mathbb R^{n\times m}$ describes where the basis vectors of $\mathbb R^3$ "land" when moved through the transformation described by the matrix $A$. The scheme that we use to describe basis vectors does not matter, only how they move with respect to the way we describe them. This final idea in discussed in this additional video by Sanderson, and is key to many application of linear algebra.