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Thread: 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 $\displaystyle T $ be a linear operator on $\displaystyle V$
    Suppose that $\displaystyle W$ is a $\displaystyle T$-invariant subspace of $\displaystyle V$, $\displaystyle dim(W)=k$
    Show that there exists a basis $\displaystyle \alpha $ for $\displaystyle V$ such that $\displaystyle [T]_{\alpha}$ has the form

    $\displaystyle \[
    \left( {\begin{array}{cc}
    A & B \\
    O & C \\
    \end{array} } \right)
    \]$

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

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

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

    So what is this question asking us to reduce the matrix to a $\displaystyle 2*2$ 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 $\displaystyle W$ and complete it to get a base of $\displaystyle V$.
    The main interest is to write $\displaystyle T$ 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 $\displaystyle T $ be a linear operator on $\displaystyle V$
    Suppose that $\displaystyle W$ is a $\displaystyle T$-invariant subspace of $\displaystyle V$, $\displaystyle dim(W)=k$
    Show that there exists a basis $\displaystyle \alpha $ for $\displaystyle V$ such that $\displaystyle [T]_{\alpha}$ has the form

    $\displaystyle \[
    \left( {\begin{array}{cc}
    A & B \\
    O & C \\
    \end{array} } \right)
    \]$

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

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

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

    So what is this question asking us to reduce the matrix to a $\displaystyle 2*2$ 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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