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Math Help - Linear Transformation and Matrix Representations

  1. #1
    Senior Member I-Think's Avatar
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    Linear Transformation and Matrix Representations

    Let V be a n-dimensional vector space, and let T be a linear operator on V
    Suppose that W is a T-invariant subspace of V, dim(W)=k
    Show that there exists a basis \alpha for V such that [T]_{\alpha} has the form

    \[<br />
\left( {\begin{array}{cc}<br />
 A & B  \\<br />
 O & C  \\<br />
 \end{array} } \right)<br />
\]

    where A is a k*k matrix and O is the (n-k)*k zero matrix

    Request
    The question itself confuses me. I am not sure what the question wants

    If \alpha is a basis for V, then \alpha has n vectors and
    [T]_{\alpha} is a n*n matrix

    So what is this question asking us to reduce the matrix to a 2*2 matrix with the entries themselves being matrices?
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  2. #2
    Super Member girdav's Avatar
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    You can choose a base for W and complete it to get a base of V.
    The main interest is to write T in a simpler way.
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  3. #3
    MHF Contributor

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    Quote Originally Posted by I-Think View Post
    Let V be a n-dimensional vector space, and let T be a linear operator on V
    Suppose that W is a T-invariant subspace of V, dim(W)=k
    Show that there exists a basis \alpha for V such that [T]_{\alpha} has the form

    \[<br />
\left( {\begin{array}{cc}<br />
 A & B  \\<br />
 O & C  \\<br />
 \end{array} } \right)<br />
\]

    where A is a k*k matrix and O is the (n-k)*k zero matrix

    Request
    The question itself confuses me. I am not sure what the question wants

    If \alpha is a basis for V, then \alpha has n vectors and
    [T]_{\alpha} is a n*n matrix

    So what is this question asking us to reduce the matrix to a 2*2 matrix with the entries themselves being matrices?
    That's one way of looking at it but it is really just saying that the matrix has the left k columns of the bottom n-k rows all 0s. Use girdav's suggestion- select an ordered basis for V such that the first k vectors form a basis for W. Remember that the matrix representation of a linear transformation, in a given basis, has columns that are the coefficients of the vector you get by applying the linear transformation to the basis vectors. Since the first k basis vectors are in W and W is T-invariant, what will those first k columns look like?
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