Linear Transformations
Chapter 2 • Section 2-6
"Functions that preserve the structure of space. 🔄"
✨ What is a Linear Transformation?
A function $T: V \to W$ between vector spaces is a linear transformation if it satisfies two key properties:
1. Preserves Addition
$T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$
2. Preserves Scaling
$T(c\vec{u}) = cT(\vec{u})$
Matrix Transformations
Every linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ is induced by an $m \times n$ matrix $A$:
The columns of $A$ are simply where the basis vectors land: $A = [T(\vec{e}_1) \dots T(\vec{e}_n)]$.
Finding the Standard Matrix
To find the matrix $A$ for a transformation $T$, just see what $T$ does to the standard basis vectors $\vec{e}_1, \vec{e}_2, \dots$.
Example
Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be a transformation that rotates points 90° counter-clockwise.
1. Transform $\vec{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$
Rotated 90°, it becomes $\begin{bmatrix} 0 \\ 1 \end{bmatrix}$.
2. Transform $\vec{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$
Rotated 90°, it becomes $\begin{bmatrix} -1 \\ 0 \end{bmatrix}$.
Thus, the matrix $A$ is:
🔧 Linear Operators & Examples
Linear Operators
A linear transformation $T: V \to V$ (from a space to itself) is called a linear operator.
Example: Rotation in $\mathbb{R}^2$ is an operator. Projection is an operator.
Differentiation ($D$)
Map $D: \calP_n \to \calP_{n-1}$ defined by $D(p(x)) = p'(x)$.
Derivative of a sum is sum of derivatives!
Integration ($I$)
Map $I: \calP_n \to \calP_{n+1}$ defined by $I(p(x)) = \int_0^x p(t)dt$.
Integrals are linear too.
Transpose ($T$)
Map $T: \M_{nn} \to \M_{nn}$ defined by $T(A) = A^T$.
$(A+B)^T = A^T + B^T$.
Theorem: Determined by Basis
A linear transformation is completely determined by its action on a basis.
If you know $T(\vec{b}_1), \dots, T(\vec{b}_n)$ for a basis, you know $T(\vec{v})$ for any vector $\vec{v}$!
2D Transformation Visualizer
See how a matrix $A$ transforms the grid and a vector $\vec{v}$.
Blue: Original Grid/Vector. Red: Transformed Grid/Vector.
Key Properties
Zero to Zero
Linear transformations always map the zero vector to the zero vector:
$T(\vec{0}) = \vec{0}$
Preserves Linear Combinations
This is the "super property" that combines addition and scaling:
$T(c_1\vec{v}_1 + \dots + c_k\vec{v}_k) = c_1T(\vec{v}_1) + \dots + c_kT(\vec{v}_k)$
Preserves Negatives
$T(-\vec{x}) = -T(\vec{x})$
🧠 Knowledge Check Win $15
Which of the following is NOT a linear transformation?
Hint: Check if $T(0) = 0$.