Chapter 8.1, Problem 19E

Consider a linear transformation L from $R^m$ to $R^n$ . Show that there exists an orthonormal basis ${\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,\mathrm{...},{\stackrel{\to }{v}}_m$ of $R^m$ such that the vectors $L\left({\stackrel{\to }{v}}_1\right),L\left({\stackrel{\to }{v}}_2\right),\mathrm{...},L\left({\stackrel{\to }{v}}_m\right)$ are orthogonal. Note that some of the vectors $L\left({\stackrel{\to }{v}}_i\right)$ may be zero. Hint: Consider an orthonormal eigenbasis ${\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2,\mathrm{...},{\stackrel{\to }{v}}_m$ for the symmetric matrix $A^{T}A$ .