Linear Transformations

Linear Transformation

A linear transformation from

V

W

is a function

T : V \to W

with the following properties for any

u, v \in V

and

c \in F

Inner Product

An inner product is a function

⟨ \cdot, \cdot ⟩ : V \times V \to F

where

V

is a vector space over the field

F

, it assigns to each pair

𝐯, 𝐰 \in V

a real number

⟨ 𝐯, 𝐰 ⟩

such that, for all

𝐮, 𝐯, 𝐰 \in V

and

α \in 𝐅

and satisfies:

Inner Product Space

An inner product space is a vector space

V

over the field

F

together with an inner product

The Dot Product in Rn Forms an Inner Product

The dot product in

ℝ^{n}

forms an inner product

Schwarz Inequality

For all

x, y \in ℝ^{n}

we have

| ⟨ x, y ⟩ | \leq ‖ x ‖ ‖ y ‖

and

| ⟨ x, y ⟩ | = ‖ x ‖ ‖ y ‖

if and only if

x, y

are collinear

The Norm Satisfies the Triangle Inequality

For any

x, y \in ℝ^{n}

we have

‖ x + y ‖ \leq ‖ x ‖ + ‖ y ‖