User:IssaRice/Linear algebra/Matrix of a linear transformation

Notation

Axler writes $\mathcal{M}(T,(v_1,\dots ,v_n),(u_1,\dots ,u_m))$. Axler never abbreviates bases, although if he did, the notation would look like $\mathcal{M}(T,\beta ,{\beta }^{\prime })$ where $\beta :=(v_1,\dots ,v_n)$ and ${\beta }^{\prime }:=(u_1,\dots ,u_m)$.

Tao writes $[T{]}_{\beta }^{{\beta }^{\prime }}$, where the "in" basis is a subscript and the "out" basis is a superscript. With vectors, $[v{]}_{\beta }$ is a row vector and $[v{]}^{\beta }$ is a column vector. This means that we can write $[T{]}_{\beta }^{{\beta }^{\prime }}[v{]}^{\beta }=[v{]}^{{\beta }^{\prime }}$ and think of the lower $\beta$ canceling out the upper $\beta$.

Treil writes $[T{]}_{{\beta }^{\prime }\beta }$ or sometimes $[T{]}_{{\beta }^{\prime },\beta }$. With vectors, we can write $[T{]}_{{\beta }^{\prime },\beta }[v{]}_{\beta }=[v{]}_{{\beta }^{\prime }}$ so that the bases "balance".