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Lecture 15

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

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  1. Lecture 15 • Projection • Least squares • Projection matrix Linear Algebra---Meiling CHEN

  2. C(P) = line through Rank(P)=1 Projection matrix Linear Algebra---Meiling CHEN

  3. Properties of projection matrix : 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

  4. Find Key: is perpendicular to plane A What subspace of in ? Ans: e is in Linear Algebra---Meiling CHEN

  5. 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

  6. Check: Least squares Fitting by a line Solve by projection Be continue….. No solution Linear Algebra---Meiling CHEN

  7. 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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