Linear transformation of an orthonormal basis

[zwingtip]

Homework Statement

Consider a linear transformation L from R^m to Rⁿ. Show that there is an orthonormal basis {v₁,...,v_m} of R^m to Rⁿ such that the vectors {L(v₁),...,L(v_m)} are orthogonal. Note that some of the vectors L(v_i) may be zero. HINT: Consider an orthonormal basis {v₁,...,v_m} for the symmetric matrix A^TA.

Homework 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

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.

 

[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?