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    Diagonal Matrix Representation : Linear Mapping, Basis and Kernels

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    A linear mapping T: R3 → R2 is defined on the standard basis vectors via:
    T (e1) = (0, 0), T (e2) = (1, 1), T (e3) = (1, -1)

    i. Calculate T(4,-1,3)
    ii. Find the dimension of the range of T and the dimension of the kernel of T.
    iii. Find the matrix representation of T relative to the standard bases in R3, R2.
    iv. Find bases {v1, v2, v3} for R3 and {w1, w2} for R2 with respect to which T has diagonal matrix representation.

    Please provide a detailed solution to all parts of the above question.

    Also, please give a little background theory regarding part 4 or at least a definition for diagonal matrix representation.
    For comparison,
    my answers to the first three parts are:

    i. (2,-4)
    ii. dim imT = 2, dim kerT = 1
    iii. the matrix below:

    0 1 1
    0 1 -1

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    https://brainmass.com/math/matrices/diagonal-matrix-representation-linear-mapping-basis-kernels-50420

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    PROBLEM

    A linear mapping T: R3 → R2 is defined on the standard basis vectors via:
    T (e1) = (0, 0), T (e2) = (1, 1), T (e3) = (1, -1)

    i. Calculate T(4,-1,3)
    ii. Find the dimension of the range of T and the dimension of the kernel of ...

    Solution Summary

    Diagonal Matrix Representation, Linear Mapping, Basis and Kernels are investigated. The solution is detailed and well presented.

    $2.19