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Romero

  • 3 years ago

Given a basis for the subpace H, if it's not an orthogonal set what can you do in order to make it an orthogonal set or is that not possible?

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  1. anonymoustwo44
    • 3 years ago
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    It is possible :D do you know the gram-schimdt process?

  2. LagrangeSon678
    • 3 years ago
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    (depending) you could also simply multiplly each vector in the base by its magnitude

  3. Romero
    • 3 years ago
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    Oh my prof skipped that chapter hahaha

  4. Romero
    • 3 years ago
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    Ill learn it later. If she didnt teach it its probably not going to be in a test. You can post a useful link if you would like I'll read it later. Thanks!

  5. anonymoustwo44
    • 3 years ago
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    it goes like this, suppose we are given a basis B={v1,v2,..,vn}. Now to make this an orthogonal set of vectors which we will call O={u1,u2,..,un}, we do the following: let u1=v1 now let W={u1} so we know that component v2 orthogonal to W(hence orthogonal to all vectors in it)=v2-orthogonal projection of v2=v2-<v2,u1>u1/norm(u1)^2 and that will give us u1 so u2=v2-<v2,u1>u1/norm(u1)^2 for u3, let W={u1,u2} so the component of v3 orthogonal to W=v3-<v3,u1>u1/norm(u1)^2-<v3,u2>u2/norm(u2)^2 continue this process till vn now you can see the pattern already right? so to construct an orthogonal basis O={u1,u2,...,un} given a basis B={v1,v2,...,vn}, we do the following: u1=v1 u2=v2-<v2,u1>u1/norm(u1)^2 u3=v3-<v3,u1>u1/norm(u1)^2-<v3,u2>u2/norm(u2)^2 continue this process till we get to: un=vn-<vn,u1>u1/norm(u1)^2-<vn,u2>u2/norm(u2)^2-...-<vn,un-1>un-1/norm(un-1)^2 only the norm is squared. the <u,v>'s are just inner products.

  6. anonymoustwo44
    • 3 years ago
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    and that will give us u2* correction :))

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