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Linear Algebra: orthogonal components of a vector

  1. May 6, 2012 #1
    1. The problem statement, all variables and given/known data
    Let V = Gen{ [0;5;1;2], [4;0;-2;1], [5,1,0,1]}. Define u11=v1. Indicate the coordinates of u2, the orthogonal component of v2 to V1=Gen{u1.


    2. Relevant equations
    I know V has to be a vector space. If there is a subspace W with an orthogonal basis B={v1,...,vk} then the orthogonal component of any vector u to W is

    uc=u - (<u,v1>/<v1,v1>)*v1 - ... - (<u,vk>/<vk,vk>)*vk


    3. The attempt at a solution
    I don-t even know where to start since the basis given is not orthogonal. Do I have to construct an orthogonal basis from the three vectors given and then proceed to solve it? I didn't see this topic in class so I didn't see the professor do this, I'm doing this for extra credit on the subject. Anyone knows where I could find a worked out example of this type of problems? Thank you.
     
  2. jcsd
  3. May 6, 2012 #2

    HallsofIvy

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    Staff Emeritus
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    Since the problem is that the basis is not orthogonal, use the "Gram-Schmidt orthogonalization procedure".
     
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