A matrix is a rectangular block of numbers. For everything that follows it is best read as a row of column vectors standing side by side, because that is how a system of equations produced it.

A timetable is a block too. Reading down a column gives one day, reading across a row gives one hour, and both readings describe the same timetable.

An m by n matrix has m rows and n columns, and the row count is always written first. The entry in row i and column j is $a_{ij}$, again row first. The same block read as n column vectors of $\mathbb{R}^m$ is the reading this chapter uses. Two matrices are equal when they have the same shape and agree in every entry.

• $A$ is $m\times n$ when it has m rows and n columns.
• $a_{ij}$ sits in row i and column j, so $a_{23}$ is row 2, column 3.

$\begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}$ is

Three rows of two numbers each, so three comes first.

in $\begin{pmatrix}1&2\\3&4\end{pmatrix}$, $a_{12}=2$ and

The two indices name different entries, so their order matters.

$\begin{pmatrix}1&2\\3&4\end{pmatrix}$ has ${\mathbf{a}}_1=($

Down the first column: 1 then 3.

One is $1\times2$ and the other $2\times1$, so the comparison stops there.