Orthogonal Sets

Chapter 5 • Section 5-3

"When vectors meet at 90 degrees, magic happens. ✨"

📐 Orthogonal & Orthonormal Sets

Definitions

  • Two vectors $\vec{u}, \vec{v}$ are orthogonal if $\vec{u} \cdot \vec{v} = 0$.
  • A set $\{\vec{u}_1, \dots, \vec{u}_k\}$ is an orthogonal set if every pair is orthogonal.
  • It is an orthonormal set if it is orthogonal AND every vector is a unit vector ($\norm{\vec{u}_i} = 1$).

Theorem

An orthogonal set of non-zero vectors is always linearly independent.

⭐ Key Properties

Cauchy-Schwarz Inequality

For any vectors $\vec{u}, \vec{v}$ in $\mathbb{R}^n$:

$$ |\vec{u} \cdot \vec{v}| \le \norm{\vec{u}} \norm{\vec{v}} $$

Triangle Inequality

For any vectors $\vec{u}, \vec{v}$ in $\mathbb{R}^n$:

$$ \norm{\vec{u} + \vec{v}} \le \norm{\vec{u}} + \norm{\vec{v}} $$

Pythagoras Theorem

Two vectors $\vec{u}, \vec{v}$ are orthogonal if and only if:

$$ \norm{\vec{u} + \vec{v}}^2 = \norm{\vec{u}}^2 + \norm{\vec{v}}^2 $$

Fourier Expansion

If $\{\vec{f}_1, \dots, \vec{f}_n\}$ is an orthogonal basis for $\mathbb{R}^n$, then any vector $\vec{x}$ can be written as:

$$ \vec{x} = \frac{\vec{x} \cdot \vec{f}_1}{\norm{\vec{f}_1}^2}\vec{f}_1 + \dots + \frac{\vec{x} \cdot \vec{f}_n}{\norm{\vec{f}_n}^2}\vec{f}_n $$

The coefficients are called Fourier coefficients. This is much easier than solving a system of equations!

Fourier Expansion Calculator (2D)

Given an orthogonal basis $\{\vec{f}_1, \vec{f}_2\}$ and a target vector $\vec{x}$, calculate the coefficients.

Must be orthogonal to $\vec{f}_1$!

🧠 Knowledge Check Win $15

If $\{\vec{u}_1, \vec{u}_2\}$ is an orthonormal set, what is $\norm{\vec{u}_1}$?