11 Linear Transformations
$$ T: \mathbb{R}^n \to \mathbb{R}^m $$Properties
A
- $\forall \vec{u}, \vec{v} \in \mathbb{R}^n, T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$
- 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}$
- Addition: $T(\vec{u} + \vec{v}) = k(\vec{u} + \vec{v}) = k\vec{u} + k\vec{v} = T(\vec{u}) + T(\vec{v}) \; \checkmark$
- Scalar Multiplication: $T(d\vec{u}) = k(d\vec{u}) = d(k\vec{u}) = dT(\vec{u}) \; \checkmark$
Matrix representation:
$$ A = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} $$