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Lecture Notes 13 - Lecture13 Lecture 13 Orthonormal Bases...

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Lecture13 1 Outline: 1) Motivation: A T Ax =A T b as a projection problem using columns of A as a basis for C(A) 2) But some bases are better than others: Orthonormal Bases A) Definition and Q matrices (Q T Q=I) B) Q matrices and Least-Squares problems B) Examples of Q matrices C) General Properties of Q Matrices D) Proof by basketball redux 3) Turning A to Q: Gram-Schmidt Othogonalization Lecture 13: Orthonormal Bases and Q matrices (towards the QR Factorization) Orthogonal Projection onto a Subspace Orthonormal bases Definition: a set of vectors q 1 ,q 2 ,...,q n in R m (m>=n) are said to be orthonormal if 1) they are all unit vectors ||q i ||=1 2) they are all mutually orthogonal: q i T q j = 0 if i j ultra compact definition q i T q j = δ ij = Orthonormal bases Theorem: any set of orthogonal vectors is linearly independent (and therefore form a basis for span(q i )) Proof: just use the fundamental definition of linear independence. the q i are linearly independent if c 1 q 1 + c 2 q 2 + ... c n q n =0 iff all c i = 0
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