4.1 Matrices as Linear Transformations

So far, we have viewed a matrix as a static spreadsheet of numbers representing data coefficients. But in geometry and deep learning, a matrix is a dynamic action!

A matrix acts like a function that takes a vector, bends/stretches the space, and outputs a new vector. We call this action a Linear Transformation.

1. What makes a transformation "Linear"?

A transformation is linear if it satisfies two strict rules:

1. The origin must stay fixed: The point $(0, 0)$ must never move.
2. Grid lines must remain straight and parallel: Grid lines cannot curve, bend, or squeeze together unevenly.

This means that any straight line in our original coordinate system must remain a straight line after the transformation.

2. The Secret: Track the Basis Vectors

It seems incredibly complicated to track where millions of points in space land after a transformation. But there is a shortcut!

Because space is uniform and grid lines remain straight and parallel, we only need to track the standard basis vectors:

• $\hat{i} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ (the unit step along the X axis).
• $\hat{j} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ (the unit step along the Y axis).

If we know where $\hat{i}$ and $\hat{j}$ land, we can calculate where any other vector in the entire space lands!

3. How a Matrix Encodes a Transformation

A $2 \times 2$ matrix is actually just a compact package storing the landing spots of $\hat{i}$ and $\hat{j}$:

$M = \begin{bmatrix} \color{blue}{a} & \color{green}{b} \\ \color{blue}{c} & \color{green}{d} \end{bmatrix}$

• Column 1 ($\begin{bmatrix} \color{blue}{a} \\ \color{blue}{c} \end{bmatrix}$): The coordinate where $\hat{i}$ lands ($\hat{i}'$).
• Column 2 ($\begin{bmatrix} \color{green}{b} \\ \color{green}{d} \end{bmatrix}$): The coordinate where $\hat{j}$ lands ($\hat{j}'$).

Multiplying a Vector

When we multiply a vector $\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}$ by matrix $M$, we are simply scaling the transformed basis vectors:

$M\mathbf{v} = x \begin{bmatrix} a \\ c \end{bmatrix} + y \begin{bmatrix} b \\ d \end{bmatrix}$

• Example: If we want to find where vector $\mathbf{v} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$ lands under transformation $M = \begin{bmatrix} 1.5 & 0.5 \\ 0.5 & 1.5 \end{bmatrix}$: $M\mathbf{v} = 2 \begin{bmatrix} 1.5 \\ 0.5 \end{bmatrix} + 3 \begin{bmatrix} 0.5 \\ 1.5 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \end{bmatrix} + \begin{bmatrix} 1.5 \\ 4.5 \end{bmatrix} = \begin{bmatrix} 4.5 \\ 5.5 \end{bmatrix}$

4. Interactive Transformation Workspace

Use the sliders to modify matrix entries $a, b, c,$ and $d$. Observe how columns of the matrix map directly to the landing coordinates of the blue basis arrow ($\hat{i}'$) and green basis arrow ($\hat{j}'$).

Try the Shear preset: Notice how the grid tilts, sliding the top of the grid to the right while keeping the bottom line fixed.

5. Check Your Understanding