Why do matrix columns describe a map?

Source scope: §3.2.1. Worked numerical examples, experiments, and practice are teaching supplements unless identified as source results.

Follow one basis direction

For $T(x,y)=(x+y,2y)$, the standard basis images are $Te_1=(1,0)$ and $Te_2=(1,2)$. Put those images in columns:

The first input coordinate weights the first column. Linearity explains why those two columns determine every output.

In general let $V,W$ have bases $v_1,\ldots,v_n$ and $w_1,\ldots,w_m$. The matrix $A\in\mathbb R^{m\times n}$ of $T:V\to W$ is defined by

Its $j$th column is the coordinate list of $Tv_j$ in the chosen output basis. Changing a basis can change the matrix without changing the underlying map. Conversely any real $m\times n$ matrix induces $T:\mathbb R^n\to\mathbb R^m$, $Tx=Ax$; its matrix in the standard bases is $A$ itself.

Turn columns into rows

The transpose has shape $n\times m$ and entries $(A^T)_{ij}=A_{ji}$. For compatible matrices and a real scalar, the source lists

The last rule reverses order. As a teaching verification, its $(i,j)$ entry is $\sum_k A_{jk}B_{ki}=\sum_k(B^T)_{ik}(A^T)_{kj}$. Both sides have shape matching the transpose of $AB$.

Transfer

If a map sends $e_1$ to $(2,-1)$ and $e_2$ to $(0,3)$, where does $(1,2)$ go?