6.2 - Orthogonal Sets

A set is orthogonal IF:

If is an orthogonal set of nonzero vectors, then the set is:

• linearly independent

• a basis for its span

So:

orthogonal nonzeroautomatically a basis

Orthogonal Basis

A basis that is also an orthogonal set

Coordinates in an Orthogonal Basis

If is an orthogonal basis for a subspace W, then any Y in W can be written as:

with

This is how you find linear combination coefficients

Orthogonal Projection

Projection of onto a single vector

proj u

Decomposition Using Two Orthogonal Vectors

If and are orthogonal

Orthonormal Sets

A set is orthonormal if

• vectors are orthogonal

• AND each vector has length

A matrix has orthonormal columns

If has orthonormal columns and

a.

b.

c.