Matrix Transformations

Chapter 2 • Section 2-2

"I'm not just a matrix... I'm a transformer! 🤖"

📐 Vectors & $Ax$

Matrix-Vector Multiplication

If $A$ has columns $\vec{a}_1, \dots, \vec{a}_n$ and $\vec{x}$ has entries $x_1, \dots, x_n$, then:

$$ A\vec{x} = x_1\vec{a}_1 + x_2\vec{a}_2 + \dots + x_n\vec{a}_n $$

It's a linear combination of the columns of $A$!

Matrix Equation

$$ A\vec{x} = \vec{b} $$

This is just a compact way to write a system of linear equations.

Dot Product Rule

Entry $i$ of $A\vec{x}$ is the dot product of Row $i$ of $A$ and $\vec{x}$.

$$ (Row_i) \cdot \vec{x} $$

🎯 Target Practice Win $10

Compute: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

Hint: It picks the first column!

Transformations

A transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ is like a function machine. It takes a vector input and gives a vector output.

Matrix Transformation

Defined by $T(\vec{x}) = A\vec{x}$.

Every matrix $A$ defines a transformation!

Rotations in $\mathbb{R}^2$

Rotation by angle $\theta$ is a linear transformation!

Rotation Matrix $R_\theta$

$$ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} $$

🔄 Spin Cycle Win $20

What is the matrix for a rotation by $90^\circ$ ($\pi/2$)?