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Lecture 15. Projection Least squares Projection matrix. C(P) = line through Rank(P)=1. Projection matrix. Properties of projection matrix :. Project twice. Why project?. Because may have no solution.

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Lecture 15
Lecture 15

  • Projection

  • Least squares

  • Projection matrix

Linear Algebra---Meiling CHEN


C(P) = line through

Rank(P)=1

Projection matrix

Linear Algebra---Meiling CHEN


Project twice

Why project?

Because may have no solution

Solve instead, where P is projection of b onto C(A)

is perpendicular to plane

Plane of

=column space of A

Linear Algebra---Meiling CHEN


Find

Key: is perpendicular to plane A

What subspace of in ?

Ans: e is in

Linear Algebra---Meiling CHEN


Projection matrix

Case 1. A is not a square matrix so this equation is not true

Case 2. A is a square matrix and invertible, means b is in C(A) than the projection is identity

Linear Algebra---Meiling CHEN


Check:

Least squares

Fitting by a line

Solve by projection

Be continue…..

No solution

Linear Algebra---Meiling CHEN


Exercise:

Let be a linear operator. For any

Where p is the projection of z on the line find

Linear Algebra---Meiling CHEN


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