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Thread: Some problem here :(

  1. #1
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    Can anyone please explain some Subspace problem here :(

    Let F: R3->R3 be the linear transformation defined by the orthogonal projection of (v belongs to R3) onto the subspace W= {(x,y,z)|x+y+z=0}
    a. Find the standard matrix A of F
    b. Show that A(A - I) = 0 Why would this be true in general (for any subspace W)?

    thanks in advance
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  2. #2
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    Quote Originally Posted by pdnhan View Post
    Let F: R3->R3 be the linear transformation defined by the orthogonal projection of (v belongs to R3) onto the subspace W= {(x,y,z)|x+y+z=0}
    a. Find the standard matrix A of F
    b. Show that A(A - I) = 0 Why would this be true in general (for any subspace W)?
    The vector normal to W is \mathbf{n}=(1,1,1). Given a vector \mathbf{v} = (x,y,z) in \mathbb{R}^3, the effect of F on \mathbf{v} will be to add a multiple of \mathbf{n} to \mathbf{v} so as to take it to the subspace W. The condition for \mathbf{v} + \lambda\mathbf{n} = (x+\lambda,y+\lambda,z+\lambda) to belong to W is x+y+z+3\lambda = 0. Solve that to see that F(\mathbf{v}) = (\tfrac23x-\tfrac13y-\tfrac13z, -\tfrac13x+\tfrac23y-\tfrac13z, -\tfrac13x-\tfrac13y+\tfrac23z). From that, you should be able to write down the matrix A and verify that A(A-I)=0.

    For the last part, if A is the matrix of the orthogonal projection onto a subspace, then IA is the matrix of the orthogonal projection onto the (orthogonal) complementary subspace.
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  3. #3
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    Smile

    cheers man
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  4. #4
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    hey man, can you please tell me how to get the value of A so I can compare, and your solution to part b) as well?
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  5. #5
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    Quote Originally Posted by pdnhan View Post
    hey man, can you please tell me how to get the value of A so I can compare, and your solution to part b) as well?
    You tell us yours first, then I'll let you know whether I agree.
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