Base change/Transformation matrix/Varia/Remark

The $j$ -th column of a transformation matrix $M_{𝔴}^{𝔳}$ consists of the coordinates of $v_{j}$ with respect to the basis $𝔴$ . The vector $v_{j}$ has the coordinate tuple $e_{j}$ with respect to the basis $𝔳$ , 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 $𝔴$ .

For a one-dimensional space and

v = c w,

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

Another important relation is

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