Linear Algebra with Applications, 10th Edition

6.5

1. 2. 

   1. (a) ${\sigma }_1=\sqrt{10},{\sigma }_2=0$;

   2. (b) ${\sigma }_1=3,{\sigma }_2=2$;

   3. (c) ${\sigma }_1=4,{\sigma }_2=2$;

   4. (d) ${\sigma }_1=3,{\sigma }_2=2,{\sigma }_3=1$. The matrices U and V are not unique. The reader may check his or her answers by multiplying out $U\mathrm{\Sigma }{V}^T$.

2. 3. 

   1. (b) rank of $A=2,A\prime =\left[\begin{array}{cc}\hfill 1.2& \hfill -2.4\\ \hfill -0.6& \hfill 1.2\end{array}\right]$

3. 4. The closest matrix of rank 2 is

   $$\left[\begin{array}{ccc}\hfill -2& \hfill 8& \hfill 20\\ \hfill 14& \hfill 19& \hfill 10\\ \hfill 0& \hfill 0& \hfill 0\end{array}\right]\mathrm{,}$$

   The closest matrix of rank 1 is

   $$\left[\begin{array}{ccc}\hfill 6& 12& \hfill 12\\ \hfill 8& \hfill 16& \hfill 16\\ \hfill 0& \hfill 0& \hfill 0\end{array}\right]$$

4. 5. 

   1. (a) basis for $R(A^T):\{{\mathbf{v}}_1={(\frac{2}{3},\frac{2}{3},\frac{1}{3})}^T,{\mathbf{v}}_2={(-\frac{2}{3},\frac{1}{3},\frac{2}{3})}^T\}$; basis for $N(A):\{{\mathbf{v}}_3={(\frac{1}{3},-\frac{2}{3},\frac{2}{3})}^T\}$

5. 6. Hint: $A^{T}A=A^2$

6. 8. Hint: $A^{T}A=V{\mathrm{\Sigma }}^T\mathrm{\Sigma }{V}^T\text{and}A{A}^T=U\mathrm{\Sigma }{\mathrm{\Sigma }}^T{U}^T$.

7. 9. Hint: $\text{det}\left(A^{T}A\right)\text{=}\text{det}(A{)}^2$.

8. 12. Hint: Plug

   $$\widehat{\mathbf{\text{x}}}=A^{+}\text{b}=U{\mathrm{\Sigma }}^{+}{V}^T\text{b}$$

   into the normal equations.