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    Matrix Representation of a Linear Operator on N-dimensional Vector Space

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    1) Let A be a linear operator on n-dimensional vector space V if {see attachment}. For some {see attachment}, prove that matrix representation {see attachment} of A with respect to a basis ...

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    https://brainmass.com/math/linear-algebra/matrix-representation-linear-operator-vector-space-38746

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    1) Let A be a linear operator on n-dimensional vector space V if A 0 and A = 0
    For some V, prove that matrix representation of A with respect to a basis
    X=(x ,x ,................x ) is similar to the .
    I have the answer but I do not understand very well
    There is the answer
    1) We have to prove that , A , A ,................ A .Is linearly independent which is easy.
    2)The book said that the eigenvalues of A are all zero. This I don't know why.
    3) The book said that the matrix of A under the basis is the matrix with all (i, i+1)-entries 1 and 0 elsewhere. This I don't understand.

    2)Prove that if and is continuous in and
    in this one we are studying inner product.

    2) Prove that if A =A than A is similar to a diagonal matrix
    I have the answer for this one too but I have some question
    There is the answer
    1. the only possible eigenvalues are 1 and 0. this I get it
    2. Theonly eigenvalues of A are 0 and 1 Hence A is similar to a diagonal matrix of the form
    D = (1,1,1,1,.....,0,0,.......0) where the number of 1's is equal to the rank of A. This I don't know why.

    I will answer your these questions below.

    For Question 1.

    1) We have to prove that , A , A ,................ A is linearly independent.
    In order to show that , A , A ...

    Solution Summary

    The matrix representation of a linear operator on n-dimensional vector space is investigated. Linear independence and eigenvalues are discussed. The solution is detailed and well presented.

    $2.49

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