by Seth Grief-Albert
{linear function, linear transform, linear map, linear mapping}
Linear transformations and matrices | Chapter 3, Essence of linear algebra
A linear function takes a vector input, and spits out a vector output.
$L:V\to W$ is called a linear function if it passes two tests:
$$ L(v_1+v_2)=L(v_1)+L(v_2)\\ L(\alpha v)=\alpha L(v) $$
$$ L(0_V)=0_W\to might\hspace{.1cm}be\hspace{.1cm}linear\\ L(0_V)\ne0_W\to not\hspace{.1cm}linear $$
$$ L(x,y)=(2x-3y,x+y,4x+5y) $$
$$ L(x,y)=\begin{bmatrix} 2\\ 1\\ 4\\ \end{bmatrix}x+\begin{bmatrix} -3\\ 1\\ 5\\ \end{bmatrix}y $$
$$ A_{L:std}=\begin{bmatrix} 2&-3\\ 1&1\\ 4&5\\ \end{bmatrix}\to first\hspace{.1cm}col=L(1,0)\hspace{.1cm}\&\hspace{.1cm}second\hspace{.1cm}col=L(0,1) $$
$$ L(x,y)=\begin{bmatrix} 2&-3\\ 1&1\\ 4&5\\ \end{bmatrix}\begin{bmatrix} x\\ y\\ \end{bmatrix} $$
The set of points for which $L(x,y,z)=0$: