Matrix Basics

A matrix is a rectangular array of numbers arranged in rows and columns. An $m \times n$ matrix has $m$ rows and $n$ columns. The entry in row $i$ and column $j$ is written $a_{ij}$.

Matrices are more than just arrays — they encode linear transformations. Every $m \times n$ matrix defines a function from $\mathbb{R}^n$ to $\mathbb{R}^m$. This is the central fact of Chapter 3.

The columns of a matrix are vectors in $\mathbb{R}^m$. We often write $A = [\mathbf{a}_1 \mid \mathbf{a}_2 \mid \cdots \mid \mathbf{a}_n]$ to emphasize the column structure.