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

Axler writes $\mathcal M(T, (v_1,\ldots, v_n), (u_1, \ldots, u_m))$. Axler never abbreviates bases, although if he did, the notation would look like $\mathcal M(T, \beta, \beta')$ where $\beta := (v_1,\ldots, v_n)$ and $\beta' := (u_1, \ldots, u_m)$.
Tao writes $[T]_\beta^{\beta'}$, 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 $[T]_\beta^{\beta'} [v]^\beta = [v]^{\beta'}$ and think of the lower $\beta$ canceling out the upper $\beta$.
Treil writes $[T]_{\beta'\beta}$ or sometimes $[T]_{\beta',\beta}$. With vectors, we can write $[T]_{\beta',\beta} [v]_\beta = [v]_{\beta'}$ so that the bases "balance".