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Homework Help: Linear transformation of an orthonormal basis

  1. Apr 26, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider a linear transformation L from Rm to Rn. Show that there is an orthonormal basis {v1,...,vm} of Rm to Rn such that the vectors {L(v1),...,L(vm)} are orthogonal. Note that some of the vectors L(vi) may be zero. HINT: Consider an orthonormal basis {v1,...,vm} for the symmetric matrix ATA.

    2. Relevant equations
    if v1 and v2 are eigenvectors of a symmetric matrix with distinct eigenvalues [tex]\lambda_1[/tex] and [tex]\lambda_2[/tex], then v1 and v2 are orthogonal

    3. The attempt at a solution
    I have no idea how to even start this problem and I've been trying for a couple of days. Can anybody give me a tip as to how attack it? Thanks.
  2. jcsd
  3. Apr 26, 2010 #2
    The matrix A^T A is diagonalizable by orthogonal matrices since it is symmetric. Therefore there exists an orthonormal basis v_1,..., v_m such that A^T A v_i = c_i v_i for some constant c_i. Now can you show that the Av_i are all orthogonal to each other?
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