linear map composition

remember that linear maps are just functions, so they can be composed

let $\text{[math]}$ be linear spaces, let $\text{[math]}$ be functions, if $\text{[math]}$ and $\text{[math]}$ are linear maps, then also the composite transformation $\text{[math]}$ is a linear map and its transformation matrix is equal to the matrix of $\text{[math]}$ multiplied by the matrix of $\text{[math]}$ correspondingly.

suppose $\text{[math]}$ and $\text{[math]}$ are linear transformations given by the matrices $\text{[math]}$ and $\text{[math]}$ respectively. then the composition $\text{[math]}$ is linear and
$\text{[math]}$ [cite:@calc_hubbard_2015 theorem 1.3.10]