Chapter 8.1, Problem 20E

Consider a linear transformation T from $R^m$ to $R^n$ , where $m\le n$ . Show that there exist an orthonormal basis ${\stackrel{\to }{v}}_1,\mathrm{...},{\stackrel{\to }{v}}_m$ of $R^m$ and an orthonormal basis ${\stackrel{\to }{w}}_1,\mathrm{...},{\stackrel{\to }{w}}_n$ of $R^n$ such that $T\left({\stackrel{\to }{v}}_i\right)$ is a scalar multiple of ${\stackrel{\to }{w}}_i$ , for $i=1,\mathrm{...},m$ . Hint: Exercise 19 is helpful.