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Math Help - Linear Map transformations (Linear Algebra)

  1. #1
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    Exclamation Linear Map transformations (Linear Algebra)

    Hello!

    I've a math problem I don't know really how to solve.
    The question follows:

    Let A be a linear map in the room and V1 and V1 transforms on themselves and V3 transforms on the null vector.
    You also know V3 is orthogonal to both V1 and V2.

    a) What is this kind of transformation called?

    b) Determine matrix A to this transformation if : V1=(1,3,10) V2=(2,2,11) V3=(?,?,?)
    **************************
    On b, I know how to get V3.
    V1xV2 = V3
    which is: 13e1+9e2-4e3
    V3 = (13,9,-4)

    Thanx for any help! I'm really stuck with this.
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  2. #2
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    Re: Linear Map transformations (Linear Algebra)

    Well, V3 is a multiple of that vector- you know the direction but not the length. But any multiple will work here, so, yes, (13, 9, -4) is good.

    This is a projection operator that maps any 3 vector onto the subspace spanned by V1 and V2.

    If you were to use V1, V2, and V3 as basis vectors you would get the matrix
    A= \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{bmatrix}.

    If you mean for the matrix in the "standard" basis, (1, 0, 0), (0, 1, 0), and (0, 0, 1), note that the matrix
    P= \begin{bmatrix}1 & 2 & 13 \\ 3 &  2 & 9\\ 10 & 11 & -4\end{bmatrix}
    having V1, V2, and V3 as columns is the "change of basis matrix"- it maps the standard basis into the new basis. Of course, then, P^{-1} goes the other way. That means the matrix product P^{-1}AP maps a vector in the standard basis to the new basis, then applies the operator A, then goes back to the standard basis- it gives the matrix for A in the standard basis.
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  3. #3
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    Re: Linear Map transformations (Linear Algebra)

    Thanx! :-)

    But I'm stuck again on c.

    c) Calculate A^2

    To calculate that, I need to do this: A^2 = S * D^2 * S^-1

    I've calculated matrix A from b):


    I need to find out the Eigenvalues.



    And now, how do i continue? I wanted to try Gauss method here, but I get huge numbers. Please help.
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  4. #4
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    Re: Linear Map transformations (Linear Algebra)

    I would use that formula for a large power of A but for A^2, why not just go ahead and multiply A by itself? Far simpler than finding eignvalues, eigenvectors, and then doing two multiplications instead of one!
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  5. #5
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    Re: Linear Map transformations (Linear Algebra)

    Quote Originally Posted by HallsofIvy View Post
    I would use that formula for a large power of A but for A^2, why not just go ahead and multiply A by itself? Far simpler than finding eignvalues, eigenvectors, and then doing two multiplications instead of one!
    Well actually, the whole c) says like this:

    Determent A^2, A^3 and A^100, compare and comment.

    Tip: Don't give up if you have a matrix A with fraction numbers. If you have the correct A-matrix, you'll see a pattern when you calculate A^2 and A^3 and then you can figure put A^100.
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