Transformation

Description

Mapping one vector from one vector space into another vector space by a function.

Function: \(f: \vec{a} \rightarrow \vec{b}\)

Domain: \(f: R_n \rightarrow R_m\)

Transformations

Linear Transformation (Linear Map)

A linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping \(\vec{v}\) to \(\vec{w}\) between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Rules:

• \(\vec{a}, \vec{b} \in \mathbb{R}^n\)
• \(T(\vec{a} + \vec{b}) = T(\vec{a}) + T(\vec{b})\)
• \(T(c \vec{a}) = c \, T(\vec{a})\)

We can always show a linear transformation via a matrix-vector product, like this: \(T(\vec{x}) = A\vec{x}\)

Rotation Transformation

\(R = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}\)

\(Rv = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \cos \theta - y \sin \theta \\ x \sin \theta + y \cos \theta \end{bmatrix}\)

Transformation Types

• Onto: If \(T(\vec{x}) = A_{m \times n} \vec{x}\) then \(Rank(A) = m\)
• One-to-One: If \(T(\vec{x}) = A_{m \times n} \vec{x}\) then \(Rank(A) = n\)
• Invertible: There should be a onto and one-to-one transformation

Operations

Addition

• \(T(u+v) = T(u) + T(v)\)
• \((T+S)(\vec{x}) = T(x) + S(x) = Ax + Bx = (A+B)x\)

Multiplication

• \(T(cu) = cT(u)\)
• \((cT)(\vec{x}) = c(T(x)) = c(Ax) = (cA)x\)