Linear Algebra Lecture 41

Linear Algebra Lecture 41

Segment VI Orthogonality and Least Squares

Gram-Schmidt Process

Construct an orthogonal basis {v1, v2} for W. Example 1 Let W = Span {x1, x2}, where Construct an orthogonal basis {v1, v2} for W.

Example 2 Then {x1, x2, x3} is linearly independent and thus, a basis for a subspace W of R4. Construct an orthogonal basis for W.

Given a basis {x1, …, xp} for a subspace W of Rn , define Theorem Given a basis {x1, …, xp} for a subspace W of Rn , define …

Continued …

Then {v1, …,vp} is an orthogonal basis for W. In addition Continued Then {v1, …,vp} is an orthogonal basis for W. In addition Span {v1, …, vk}= Span {x1,…, xk} for

Orthonormal Bases

Example 3

Theorem If A is an m x n matrix with linearly independent columns, then A can be factored as A = QR …

Continued Where Q is an m x n matrix whose columns form an orthonormal basis for Col A and R is an n x n upper triangular invertible matrix with positive entries on its diagonal.

Proof of the Theorem

Find a QR factorization of Example 4 Find a QR factorization of

Construct an orthonormal basis for W. Example 5 Let W = Span {x1, x2}, where Construct an orthonormal basis for W.

Given a basis {x1, …, xp} for a subspace W of Rn , define Theorem Given a basis {x1, …, xp} for a subspace W of Rn , define …

Continued …

Then {v1, …,vp} is an orthogonal basis for W. In addition Continued Then {v1, …,vp} is an orthogonal basis for W. In addition Span {v1, …, vk}= Span {x1,…, xk} for

Decomposition Theorem The Orthogonal Decomposition Theorem Let W be a subspace of Rn. Then each y in Rn can be written uniquely in the form where is in W and z is in . …

In fact, if {u1, …, up} is any orthogonal basis of W, then Continued In fact, if {u1, …, up} is any orthogonal basis of W, then and z = y – . The vector is called the orthogonal projection of y onto W and often is written as projw y.

Best Approximation Theorem Let W be a subspace of Rn, y any vector in Rn, and the orthogonal projection of y onto W. Then is the closest point in W to y, in the sense that for all v in W distinct from . …

Continued The vector in this theorem is called the best approximation to y by elements of W.

Theorem

Linear Algebra Lecture 41