Lesson 5.0

Vector
Norms

Transitioning from synthetic to analytic geometry: Similarity and the Inner Product Space.

∥v∥² = v · v ⟨u, v⟩ = 0 ℝⁿ SPACE

01

Algebraic Foundations

In an inner product space \( V \), the norm of a vector \( \vec{v} \) is defined by similarity to the dot product.

\( \|\vec{v}\| = \sqrt{\langle\vec{v}, \vec{v}\rangle} \)

This algebraic definition encapsulates the geometric "length" of the vector.

Orthogonality

Two vectors \( \vec{u} \) and \( \vec{v} \) are orthogonal if:

\( \langle\vec{u}, \vec{v}\rangle = 0 \)

02

Similarity Transformations

A Similarity Transformation is a linear operator \( T \) such that for all \( \vec{u}, \vec{v} \):

\( \|T\vec{v}\| = k \|\vec{v}\| \)

Where \( k > 0 \) is the scale factor. This preserves angles and ratios of lengths.

Theorem Interaction

The Pythagorean Theorem in \( \mathbb{R}^n \) is equivalent to the statement:

\( \|\vec{u} + \vec{v}\|^2 = \|\vec{u}\|^2 + \|\vec{v}\|^2 \)
if and only if \( \langle\vec{u}, \vec{v}\rangle = 0 \).

The Unified Theory

Whether viewed through synthetic similarity (Euclid), area scaling (Einstein), or vector norms (Hilbert), the Pythagorean relationship is the fundamental invariant of Euclidean space.

"Geometry is just linear algebra without the coordinates."

Vector Norms

Vector Norms Slides

Vector Norms Problems