Orthogonal sets

Orthogonal Sets and Basis
• Each pair of vector is orthogonal
• Linear independent → Form a Basis
• Calculate weights with Formula
Orthonormal Sets and Basis
• Is an orthogonal set
• Each vector is a unit vector
• Linear independent → Form a Basis
Matrix $U$ with Orthonormal columns and Properties
• $U^T U=I$
• 3 Properties of Matrix $U$
Orthogonal Projection and Component
• Orthogonal Projection of $y$ onto $v$
• The component of $y$ orthogonal to $v$

Orthogonal Sets and Basis
Is this an orthogonal set?
Verify that is an orthogonal basis for $\Bbb{R}^2$ , and then express as a linear combination of the set of vectors in $B$ .
Orthonormal Sets/Basis
Is set $B$ is an orthonormal basis for $\Bbb{R}^3$ ?
Let where $U$ has orthonormal columns and $U^TU=I$ . Verify that
$(Ux)\cdot (Uy)=x\cdot y$
Orthogonal Projection
Let and . Write $y$ as the sum of two orthogonal vectors, one in Span{ $v$ } and one orthogonal to $v$ .