A key insight in linear algebra is viewing a matrix not just as a table of numbers, but as a function that transforms space. Garrity emphasizes vector spaces early on. Here, we visualize $\mathbb{R}^2$. Consider a $2 \times 2$ matrix $M$:
$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$
This matrix transforms every vector $(x, y)$ in the plane to a new vector $(ax + by, cx + dy)$. We can visualize this by tracking where the standard basis vectors $\color{#b13e3e}{\mathbf{i} = (1, 0)}$ and $\color{#3e3eb1}{\mathbf{j} = (0, 1)}$ land.
Things to notice
- Linearity: Grid lines remain parallel and evenly spaced.
- Determinant: The area of the unit square (formed by $\mathbf{i}$ and $\mathbf{j}$) scales by the determinant $(ad - bc)$.
- Scaling: Try $\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ to double the size of the grid.
- Shear vs Rotation: Try $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ for a horizontal shear (x changes based on y).
- Rotation: Try $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ for a 90° counter-clockwise rotation.
- Reflection: Try $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ to reflect across the y-axis.
- Singular: Try $\begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix}$. The determinant is 0, squashing the plane into a line.