Base change/Transformation matrix/Varia/Remark

The ${}j$-th column of a transformation matrix ${}M_{\mathfrak {w}}^{\mathfrak {v}}$ consists of the coordinates of ${}v_{j}$ with respect to the basis ${}{\mathfrak {w}}$. The vector ${}v_{j}$ has the coordinate tuple ${}e_{j}$ with respect to the basis ${}{\mathfrak {v}}$, and when we apply the matrix to ${}e_{j}$, we get the ${}j$-th column of the matrix, and this is just the coordinate tuple of ${}v_{j}$ with respect to the basis ${}{\mathfrak {w}}$.

For a one-dimensional space and

${}v=cw\,,$

we have ${}M_{\mathfrak {w}}^{\mathfrak {v}}=c={\frac {v}{w}}$, where the fraction is well-defined. This might help in memorizing the order of the bases in this notation.

Another important relation is

${}{\mathfrak {v}}={{\left(M_{\mathfrak {w}}^{\mathfrak {v}}\right)}^{\text{tr}}}{\mathfrak {w}}\,.$

Note that here, the matrix is not applied to an ${}n$-tuple of ${}K$ but to an ${}n$-tuple of ${}V$, yielding a new ${}n$-tuple of ${}V$. This equation might be an argument to define the transformation matrix the other way around; however, we consider the behavior in fact as decisive.

In case

${}V=K^{n}\,,$

if ${}{\mathfrak {e}}$ is the standard basis, and ${}{\mathfrak {v}}$ some further basis, we obtain the transformation matrix ${}M_{\mathfrak {v}}^{\mathfrak {e}}$ of the base change from ${}{\mathfrak {e}}$ to ${}{\mathfrak {v}}$ by expressing each ${}e_{j}$ as a linear combination of the basis vectors ${}v_{1},\ldots ,v_{n}$, and writing down the corresponding tuples as columns. The inverse transformation matrix, ${}M_{\mathfrak {e}}^{\mathfrak {v}}$, consists simply in ${}v_{1},\ldots ,v_{n}$, written as columns.