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la_m11_handouts

# la_m11_handouts - MAC 2103 Module 11 lnner Product Spaces...

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1 1 MAC 2103 Module 11 lnner Product Spaces II 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Construct an orthonormal set of vectors from an orthogonal set of vectors. 2. Find the coordinate vector with respect to a given orthonormal basis. 3. Construct an orthogonal basis from a nonstandard basis in ℜⁿ using the Gram-Schmidt process. 4. Find the least squares solution to a linear system A x = b . 5. Find the orthogonal projection on col(A). 6. Obtain the best approximation. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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2 3 Rev.09 General Vector Spaces II http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Orthogonal Bases, Gram-Schmidt Process, Orthogonal Bases, Gram-Schmidt Process, Least Squares Least Squares and and Best Approximation Best Approximation The major topics in this module: 4 Rev.F09 How to Construct an Orthonormal Set of Vectors from an Orthogonal Set of Vectors? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. We have learned from the previous module that two vectors u and v in an inner product space V are orthogonal to each other iff < u , v > = 0. To obtain an orthonormal set, we will normalize each of the vectors in the orthogonal set. How to normalize the vectors? This can be done by dividing each of them by their respective norm and making each of them a unit vector.
3 5 Rev.F09 How to Construct an Orthonormal Set of Vectors from an Orthogonal Set of Vectors? (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Example 1: Find the orthonormal set of vectors from the following set of vectors: Let S ={ v 1 , v 2 } where v 1 = (5,0) and v 2 = (0,-3). Step 1: Verify that the set of vectors are mutually orthogonal with respect to the Euclidean inner product on ² . Step 2: Find the norm for both vectors. ! ! v 1 , ! v 2 " = ! v 1 # ! v 2 = (5)(0) + (0)( \$ 3) = 0 S is an orthogonal set . ! v 1 = 5 2 + 0 2 = 5, and ! v 2 = 0 2 + ( ! 3) 2 = 3. 6 Rev.F09 How to Construct an Orthonormal Set of Vectors from an Orthogonal Set of Vectors? (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Step 3: Normalize the vectors in the orthogonal set. Step 4: Verify that the set S is orthonormal by showing that and ! q 1 = ! v 1 ! v 1 = 5 5 , 0 5 ! " # \$ % & = (1,0), ! q 2 = ! v 2 ! v 2 = 0 3 , ' 3 3 ! " # \$ % & = (0, ' 1) ! q 1 = ! q 2 = 1. ! ! q 1 , ! q 2 " = 0 ! ! q 1 , ! q 2 " = ! q 1 # ! q 2 = (1)(0) + (0)( \$ 1) = 0, ! q 1 = ! ! q 1 , ! q 1 " 1 2 = ( ! q 1 # ! q 1 ) 1 2 = 1 2 + 0 2 = 1, and ! q 2 = ! ! q 2 , ! q 2 " 1 2 = ( ! q 2 # ! q 2 ) 1 2 = 0 2 + ( \$ 1) 2 = 1.

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4 7 Rev.F09 Orthonormal Set, Orthonormal Basis, and Orthogonal Basis http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.
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