transformation matrix

suppose that $\text{[math]}$ and $\text{[math]}$ are finite-dimensional vector spaces with ordered bases $\text{[math]}$ and $\text{[math]}$ , respectively. let $\text{[math]}$ be linear. then for each $\text{[math]}$ , there exist unique scalars $\text{[math]}$ , such that
$\text{[math]}$ we call the $\text{[math]}$ matrix $\text{[math]}$ defined by $\text{[math]}$ the matrix representation of $\text{[math]}$ in the ordered bases $\text{[math]}$ and $\text{[math]}$ and write $\text{[math]}$ . if $\text{[math]}$ and $\text{[math]}$ , then we write $\text{[math]}$
[cite:;taken from @algebra_insel_2019 chapter 2 linear transformations and matrices]

notice that the $\text{[math]}$ th column of $\text{[math]}$ is simply $\text{[math]}$ . also observe that if $\text{[math]}$ is a linear transformation such that $\text{[math]}$ , then $\text{[math]}$ .
[cite:;taken from @algebra_insel_2019 chapter 2 linear transformations and matrices]

let $\text{[math]}$ , and $\text{[math]}$ be finite-dimensional vector spaces with ordered basis $\text{[math]}$ , and $\text{[math]}$ , respectively. let $\text{[math]}$ and $\text{[math]}$ be linear transformations. then
$\text{[math]}$ [cite:;taken from @algebra_insel_2019 theorem 2.11]

the following theorem states that if we want to find the coordinates of the transformed vector $\text{[math]}$ in the basis $\text{[math]}$ , we can do so by first taking the coordinates of the original vector $\text{[math]}$ in the basis $\text{[math]}$ and then multiplying them by the matrix that represents the transformation $\text{[math]}$ with respect to those bases.

let $\text{[math]}$ and $\text{[math]}$ be finite-dimensional vector spaces having ordered bases $\text{[math]}$ and $\text{[math]}$ , respectively, and let $\text{[math]}$ be linear. then, for each $\text{[math]}$ , we have
$\text{[math]}$ [cite:;taken from @algebra_insel_2019 theorem 2.14]

let $\text{[math]}$ such that
$\text{[math]}$ $\text{[math]}$ is given by
$\text{[math]}$ more generally, we want to have $\text{[math]}$ such that
$\text{[math]}$ we have
$\text{[math]}$ and
$\text{[math]}$ which means
$\text{[math]}$

when the transformation matrix is from the basis onto itself, we may denote it by $\text{[math]}$ . we have $\text{[math]}$

the relation between $\text{[math]}$ and $\text{[math]}$ :
$\text{[math]}$ which means
$\text{[math]}$ then
$\text{[math]}$

let $\text{[math]}$ that is defined by $\text{[math]}$ , let $\text{[math]}$ and $\text{[math]}$ be 2 bases of $\text{[math]}$ .

find $\text{[math]}$ ,
find $\text{[math]}$ .

solution steps:

change-of-basis matrices ( $\text{[math]}$ , $\text{[math]}$ ): these are the tools for translating between bases.
single-basis transformation matrices ( $\text{[math]}$ , $\text{[math]}$ ): these represent the transformation $\text{[math]}$ if you work exclusively in one basis.
mixed-basis transformation matrices ( $\text{[math]}$ , $\text{[math]}$ ): these represent $\text{[math]}$ when moving from one basis to another.

the solution:

change-of-basis matrices first, we find the matrix $\text{[math]}$ which converts coordinates from basis $\text{[math]}$ to basis $\text{[math]}$ . its columns are the coordinate vectors of $\text{[math]}$ expressed in basis $\text{[math]}$ . $\text{[math]}$ this gives the change-of-basis matrix from $\text{[math]}$ to $\text{[math]}$ : $\text{[math]}$ the change-of-basis matrix from $\text{[math]}$ to $\text{[math]}$ is its inverse: $\text{[math]}$

single-basis transformation matrices next, we find $\text{[math]}$ . we apply $\text{[math]}$ to each vector in $\text{[math]}$ and find the coordinates of the result relative to $\text{[math]}$ . $\text{[math]}$ assembling the columns gives $\text{[math]}$ : $\text{[math]}$ now we find $\text{[math]}$ using the change-of-basis formula $\text{[math]}$ : $\text{[math]}$

mixed-basis transformation matrices the matrix $\text{[math]}$ takes a vector in $\text{[math]}$ , applies $\text{[math]}$ , and gives the result in $\text{[math]}$ . we find it by first transforming in $\text{[math]}$ ( $\text{[math]}$ ), then translating the result to $\text{[math]}$ ( $\text{[math]}$ ). $\text{[math]}$ similarly, $\text{[math]}$ takes a vector in $\text{[math]}$ , applies $\text{[math]}$ , and gives the result in $\text{[math]}$ . we transform in $\text{[math]}$ ( $\text{[math]}$ ), then translate to $\text{[math]}$ ( $\text{[math]}$ ). $\text{[math]}$

let $\text{[math]}$ be a linear operator on a finite-dimensional vector space $\text{[math]}$ , and let $\text{[math]}$ and $\text{[math]}$ be ordered bases for $\text{[math]}$ . suppose that $\text{[math]}$ is the change of coordinate matrix that changes $\text{[math]}$ -coordinates
into $\text{[math]}$ -coordinates. then
$\text{[math]}$ [cite:;taken from @algebra_insel_2019 theorem 2.23]