# Orthogonal sets

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##### Intros
###### Lessons
1. Orthogonal Sets Overview:
2. Orthogonal Sets and Basis
• Each pair of vector is orthogonal
• Linear independent → Form a Basis
• Calculate weights with Formula
3. Orthonormal Sets and Basis
• Is an orthogonal set
• Each vector is a unit vector
• Linear independent → Form a Basis
4. Matrix $U$ with Orthonormal columns and Properties
$U^T U=I$
• 3 Properties of Matrix $U$
5. Orthogonal Projection and Component
• Orthogonal Projection of $y$ onto $v$
• The component of $y$ orthogonal to $v$
##### Examples
###### Lessons
1. Orthogonal Sets and Basis
Is this an orthogonal set?
1. Verify that is an orthogonal basis for $\Bbb{R}^2$, and then express as a linear combination of the set of vectors in $B$.
1. Orthonormal Sets/Basis
Is set $B$ is an orthonormal basis for $\Bbb{R}^3$?
1. Let where $U$ has orthonormal columns and $U^TU=I$. Verify that
$(Ux)\cdot (Uy)=x\cdot y$
1. Orthogonal Projection
Let and . Write $y$ as the sum of two orthogonal vectors, one in Span{$v$} and one orthogonal to $v$.