Homework Help: Linear transformation of an orthonormal basis

1. Apr 26, 2010

zwingtip

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 $$\lambda_1$$ and $$\lambda_2$$, 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. Apr 26, 2010

eok20

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?