Introduction to Linear Transformations

Overview

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. Linear transformations are the functions that "play well" with linear algebra.

Definition

A function $T: V \to W$ is a linear transformation if for all $\mathbf{u}, \mathbf{v} \in V$ and scalar $c$:

1. Additivity: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
2. Homogeneity: $T(c\mathbf{v}) = cT(\mathbf{v})$

Equivalently (combined):

$T(c\mathbf{u} + d\mathbf{v}) = cT(\mathbf{u}) + dT(\mathbf{v})$

Basic Examples

Scaling

$T(\mathbf{v}) = k\mathbf{v}$ for scalar $k$

$T\begin{bmatrix} x \\ y \end{bmatrix} = k\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} kx \\ ky \end{bmatrix}$

Rotation (2D)

Rotate by angle $\theta$:

$T\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x\cos\theta - y\sin\theta \\ x\sin\theta + y\cos\theta \end{bmatrix}$

Projection

Project onto x-axis:

$T\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ 0 \end{bmatrix}$

Reflection

Reflect across x-axis:

$T\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ -y \end{bmatrix}$

Verifying Linearity

To prove $T$ is linear, verify both conditions:

Example

$T(x, y) = (2x + y, x - y)$

Check additivity:

$T(\mathbf{u} + \mathbf{v}) = T(x_1+x_2, y_1+y_2) = (2(x_1+x_2) + (y_1+y_2), (x_1+x_2) - (y_1+y_2))$

$= (2x_1+y_1, x_1-y_1) + (2x_2+y_2, x_2-y_2) = T(\mathbf{u}) + T(\mathbf{v}) \checkmark$

$T(c\mathbf{v}) = T(cx, cy) = (2cx+cy, cx-cy) = c(2x+y, x-y) = cT(\mathbf{v}) \checkmark$

$T$ is linear.

Non-Linear Examples

These are NOT linear transformations:

• $T(x) = x + 1$ (translation)
• $T(x, y) = xy$ (multiplication)
• $T(x) = x^2$ (squaring)
• $T(x) = \lvert x \rvert$ (absolute value)

Why Not Linear

$T(x) = x + 1$:

$T(0) = 1 \neq 0$

But linear transformations must satisfy $T(\mathbf{0}) = \mathbf{0}$

Properties of Linear Transformations

Property	Formula
Zero maps to zero	$T(\mathbf{0}) = \mathbf{0}$
Negation	$T(-\mathbf{v}) = -T(\mathbf{v})$
Linear combination	$T(\sum c_i\mathbf{v}_i) = \sum c_i T(\mathbf{v}_i)$

Standard Linear Transformations

Zero Transformation

$T(\mathbf{v}) = \mathbf{0}$ for all $\mathbf{v}$

Identity Transformation

$T(\mathbf{v}) = \mathbf{v}$ for all $\mathbf{v}$

Differentiation

$D: P_n \to P_{n-1}$

$D(p(x)) = p'(x)$

Integration

$\int: P_n \to P_{n+1}$

$\int(p(x)) = \int p(x) \, dx$

Composition

If $S$ and $T$ are linear, so is $S \circ T$:

$(S \circ T)(\mathbf{v}) = S(T(\mathbf{v}))$

Inverse

If $T$ is invertible (bijective), $T^{-1}$ is also linear:

$T^{-1}(T(\mathbf{v})) = \mathbf{v}$

Connection to Matrices

Every linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be represented by an $m \times n$ matrix $A$:

$T(\mathbf{v}) = A\mathbf{v}$

This is the fundamental link between transformations and matrices.