Science:Math Exam Resources/Courses/MATH307/April 2013/Question Section 201 07 (b)/Solution 1

The equation for the singular value matrix decomposition is

$A$ = $U\mathrm{\Sigma }{V}^{*}$ = $U\mathrm{\Sigma }{V}^T$

$=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}{\sigma }_1& 0& 0\\ 0& {\sigma }_2& 0\end{array}\right]\left[\begin{array}{ccc}\frac{1}{\sqrt{(}2)}& 0& -\frac{1}{\sqrt{(}2)}\\ \frac{1}{\sqrt{(}2)}& 0& \frac{1}{\sqrt{(}2)}\\ 0& 1& 0\end{array}\right]$

It can be seen that the third column of V is an orthogonal basis for N(A), and both columns of U form the basis for R(A).