Lect3-05-web

Lect3-05-web - MATH 304, Fall 2011 Linear Algebra Homework...

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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #10 (due Thursday November, 17) All problems are from Leon’s book (8th edition). Section 5.5: 6c, 29a, 29b, 30a, 30b Section 5.6: 1a, 3, 4, 7, 8 MATH 304 Linear Algebra Lecture 21: The Gram-Schmidt orthogonalization process. Eigenvalues and eigenvectors of a matrix. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1 , v 2 , . . . , v k ∈ V form an orthogonal set if they are orthogonal to each other: ( v i , v j ) = 0 for i negationslash = j . If, in addition, all vectors are of unit norm, bardbl v i bardbl = 1, then v 1 , v 2 , . . . , v k is called an orthonormal set . Theorem Any orthogonal set is linearly independent. Orthogonal projection Theorem Let V be an inner product space and V be a finite-dimensional subspace of V . Then any vector x ∈ V is uniquely represented as x = p + o , where p ∈ V and o ⊥ V . The component p is the orthogonal projection of the vector x onto the subspace V . The distance from x to the subspace V is bardbl o bardbl . If v 1 , v 2 , . . . , v n is an orthogonal basis for V then p = ( x , v 1 ) ( v 1 , v 1 ) v 1 + ( x , v 2 ) ( v 2 , v 2 ) v 2 + ··· + ( x , v n ) ( v n , v n ) v n . V o p x The Gram-Schmidt orthogonalization process Let V be a vector space with an inner product. Suppose x 1 , x 2 , . . . , x n is a basis for V . Let v 1 = x 1 , v 2 = x 2 − ( x 2 , v 1 ) ( v 1 , v 1 ) v 1 , v 3 = x 3 − ( x 3 , v 1 ) ( v 1 , v 1 ) v 1 − ( x 3 , v 2 ) ( v 2 , v 2 ) v 2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v n = x n − ( x n , v 1 ) ( v 1 , v 1 ) v 1 − ··· − ( x n , v n − 1 ) ( v n − 1 , v n − 1 ) v n − 1 . Then v 1 , v 2 , . . . , v n is an orthogonal basis for V . Span( v 1 , v 2 ) = Span( x 1 , x 2 ) v 3 p 3 x 3 Any basis x 1 , x 2 , . . . , x n −→ Orthogonal basis v 1 , v 2 , . . . , v n Properties of the Gram-Schmidt process: • v k = x k − ( α 1 x 1 + ··· + α k − 1 x k − 1 ), 1 ≤ k ≤ n ; • the span of v 1 , . . . , v k is the same as the span of x 1 , . . . , x k ; • v k is orthogonal to x 1 , . . . , x k − 1 ; • v k = x k − p k , where p k is the orthogonal projection of the vector x k on the subspace spanned by x 1 , . . . , x k − 1 ; • bardbl v k bardbl is the distance from x k to the subspace spanned by x 1 , . . . , x k − 1 . Normalization Let V be a vector space with an inner product....
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This note was uploaded on 03/19/2012 for the course STAT 211 taught by Professor Parzen during the Spring '07 term at Texas A&M.

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Lect3-05-web - MATH 304, Fall 2011 Linear Algebra Homework...

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