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Math Help - Finding matrix using linear transformations

  1. #1
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    Finding matrix using linear transformations

    Let B = (v1, v2, v3, v4) be a basis for a vector space V. Find the matrix with respect to B of the linear operator T: V -----> V defined by:

    T(v1) = (v2), T(v2) = v3, T(v3) = v4, T(v4) = v1
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  2. #2
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    Quote Originally Posted by chadlyter View Post
    Let B = (v1, v2, v3, v4) be a basis for a vector space V. Find the matrix with respect to B of the linear operator T: V -----> V defined by:

    T(v1) = (v2), T(v2) = v3, T(v3) = v4, T(v4) = v1
    When you are transforming basis e_i vectors the matrix will be

    T=\begin{bmatrix}T(e_1) && T(e_2) && T(e_3) && T(e_4) \end{bmatrix}

    Or in your case

    T=<br />
\begin{bmatrix}T(v_1) && T(v_2) && T(v_3) && T(v_4) \end{bmatrix}=<br />
\begin{bmatrix}v_2 && v_3 && v_4 && v_1 \end{bmatrix}
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  3. #3
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    Quote Originally Posted by chadlyter View Post
    Let B = (v1, v2, v3, v4) be a basis for a vector space V. Find the matrix with respect to B of the linear operator T: V -----> V defined by:

    T(v1) = (v2), T(v2) = v3, T(v3) = v4, T(v4) = v1
    The j'th column of the matrix will consist of the coefficients of T(v_j) when expressed as a linear combination of the four basis vectors.

    For example, Tv_1=v_2=0v_1+1v_2+0v_3+0v_4, so the first column of the matrix will consist of 0,1,0,0.

    Thus the matrix will be \begin{bmatrix}0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\  end{bmatrix}.
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