Base change/Transformation matrix/Varia/Remark

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

For a one-dimensional space and

${\displaystyle {}v=cw\,,}$

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

Another important relation is

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

Note that here, the matrix is not applied to an ${\displaystyle {}n}$-tuple of ${\displaystyle {}K}$ but to an ${\displaystyle {}n}$-tuple of ${\displaystyle {}V}$, yielding a new ${\displaystyle {}n}$-tuple of ${\displaystyle {}V}$. 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

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

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