11 Linear Transformations

$$ T: \mathbb{R}^n \to \mathbb{R}^m $$
A function/transformation that maps an input from one vector space into another.

Properties

A linear transformation is a type of function that satisfies 2 properties:

$\forall \vec{u}, \vec{v} \in \mathbb{R}^n, T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$
Respects vector addition
$\forall \vec{u} \in \mathbb{R}, \forall c \in \mathbb{R}, T(c\vec{u}) = cT(\vec{u})$
Respects scalar multiplication

Matrix Representation

Every linear transformation $T$ can be represented as a matrix, such that $T(\vec{v}) = A\vec{v}$ for all $\vec{v} \in \mathbb{R}^n$. $A$ is determined by how $T$ maps a basis of $\mathbb{R}^n$ to a basis of $\mathbb{R}^m$.

Examples

Dilation and contraction

Define $T: \mathbb{R}^2 \to \mathbb{R}^2$ as a dilation by a factor $k$:

$$ T(\vec{v}) = k\vec{v} \text{ for } \vec{v} = \begin{bmatrix} x \\ y \end{bmatrix} \in \mathbb{R}^2. $$

$T(\vec{u}) = k\vec{u}$

Matrix representation:

$$ A = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} $$
Points (0,1) and (1,0) dilated by a factor of 2.

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