Linear Algebra with Applications, 10th Edition

4.2

1. 1. 

   1. (a) $\left[\begin{array}{cc}\hfill -1& 0\\ \hfill 0& 1\end{array}\right]$;

   2. (c) $\left[\begin{array}{cc}0& 1\\ \hfill 1& 0\end{array}\right]$;

   3. (d) $\left[\begin{array}{cc}\frac{1}{2}& 0\\ 0& \frac{1}{2}\end{array}\right]$;

   4. (e) $\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

2. 2. 

   1. (a) $\left[\begin{array}{ccc}1& 1& 0\\ 0& 0& 0\end{array}\right]$;

   2. (b) $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$;

   3. (c) $\left[\begin{array}{ccc}\hfill -1& \hfill 1& 0\\ \hfill 0& \hfill -1& 1\end{array}\right]$

3. 3. 

   1. (a) $\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$;

   2. (b) $\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$;

   3. (c) $\left[\begin{array}{ccc}0& 0& \hfill 2\\ 3& 1& \hfill 0\\ 2& 0& \hfill -1\end{array}\right]$

4. 4. 

   1. (a) ${(0,0,0)}^T$;

   2. (b) ${(2,-1,-1)}^T$;

   3. (c) ${(-15,9,6)}^T$

5. 5. 

   1. (a) $\left[\begin{array}{cc}\hfill {\displaystyle \frac{1}{\sqrt{2}}}& {\displaystyle \frac{1}{\sqrt{2}}}\\ \hfill -{\displaystyle \frac{1}{\sqrt{2}}}& \frac{1}{\sqrt{2}}\end{array}\right]$;

   2. (b) $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

   3. (c) $\left[\begin{array}{cc}\sqrt{3}& \hfill -1\\ 1& \hfill \sqrt{3}\end{array}\right]$;

   4. (d) $\left[\begin{array}{cc}0& \hfill 1\\ 0& \hfill 0\end{array}\right]$

6. 6. $\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\end{array}\right]$

7. 7. Hint: In part (a) we must find a vector $\mathbf{c}={\left(c_1,c_2,c_3\right)}^T$ such that

   $$\mathit{⌶}\left({\mathbf{\text{e}}}_1\right)=c_1\mathit{}{\mathbf{\text{y}}}_1+c_2{\mathbf{\text{y}}}_2+c_3{\mathbf{\text{y}}}_3$$

   Since ⌶ is the identity operator, $\mathit{⌶}\left({\mathbf{e}}_1\right)={\mathbf{e}}_1,$ and hence we can rewrite the equation in the form

   $$Y\mathbf{\text{c}}={\mathbf{e}}_1$$

   It follows that

   $$\mathbf{c}=Y^{-1}{\mathbf{e}}_1$$

   Note that c is the first column of $Y^{-1}$ which is the transition matrix from the standard basis $\left[{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\right]\text{ to }\left[{\mathbf{y}}_1,{\mathbf{y}}_2,{\mathbf{y}}_3\right]\mathrm{.}$

8. 8. 

   1. (a) $\left[\begin{array}{ccc}1& \hfill 1& \hfill 1\\ 2& \hfill 0& \hfill 1\\ 0& \hfill -2& \hfill -1\end{array}\right]$;

   2. (b)  

      1. $7{\mathbf{y}}_1+6{\mathbf{y}}_2-8{\mathbf{y}}_3$,

      2. $3{\mathbf{y}}_1+3{\mathbf{y}}_2-3{\mathbf{y}}_3$,

      3. ${\mathbf{y}}_1+5{\mathbf{y}}_2-3{\mathbf{y}}_3$

9. 9. 

   1. (a) square;

   2. (b)  

      1. contraction by a factor $\frac{1}{2}$,