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

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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".