Linear Algebra: orthogonal components of a vector

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Homework Statement


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.


Homework 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


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.
 

Answers and Replies

  • #2
HallsofIvy
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Since the problem is that the basis is not orthogonal, use the "Gram-Schmidt orthogonalization procedure".
 

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