Day 23 Outline

Tuesday, April 24, 2018

6.2 A (Discrete) orthogonal set

NOTE: An orthogonal set is allowed to contain the 0 vector, so orthogonal set is not necessarily LI, but an orthogonal set of nonzero vectors is LI. Careful - this distinction comes up several times in Homework 10.

6.3 Orthogonal projections

Orthog proj along a single vector: proju y

Orthogonal decomposition theorem. W a subspace of Rn. Each y in Rn can be uniquely expressed as y = yhat + z where yhat is in W and z is in Wperp.

Specifally, if u1,...,up an orthogonal basis for W, yhat = ...

Example

projW y

Properties of orthog projections

1. projWy = y for y in W

2. Best approx theorem: y^-y closer to y than any other point v in W

Proof using Pythagoras

Example: find closest point in W (2 orthogonal vectors in R3) to y. (Student-supplied)

Theorem 10: Simpler formula for projW y in case of orthonormal basis and matrix UUT for projection.

6.4 Gram-Schmidt process

Usefulness of orthog bases - how to make?

GS process

Orthonormal basis from orthogonal basis.

QR factorization of mxn A of rank n

6.5 Least squares problems