{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
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.
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
4 7 Rev.F09 Orthonormal Set, Orthonormal Basis, and Orthogonal Basis http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.
Image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern