Linear Algebra with Applications, 10th Edition

Section 5.3 Exercises

1. Find the least squares solution of each of the following systems:

   1. $\begin{array}{cc}\hfill x_1+x_2& =3\hfill \\ \hfill 2x_1-3x_2& =1\hfill \\ \hfill 0x_1+0x_2& =2\hfill \end{array}$

   2. $\begin{array}{cc}\hfill -x_1+x_2& =10\hfill \\ \hfill 2x_1+x_2& =5\hfill \\ \hfill x_1-2x_2& =20\hfill \end{array}$

   3. $\begin{array}{cc}\hfill x_1+x_2+x_3& =4\hfill \\ \hfill -x_1+x_2+x_3& =0\hfill \\ \hfill -x_2+x_3& =1\hfill \\ \hfill x_1+x_3& =2\hfill \end{array}$

2. For each of your solutions $\hat{\mathbf{x}}$ in Exercise 1:

   1. determine the projection $\mathbf{\text{p}}=A\hat{\mathbf{x}}$.

   2. calculate the residual $r\left(\hat{\mathbf{x}}\right)$.

   3. verify that $r\left(\hat{\mathbf{x}}\right)\in N\left(A^T\right)$.

3. For each of the following systems $A\mathbf{x}=\mathbf{b}$, find all least squares solutions:

   1. $A=\left[\begin{array}{cc}1& 2\\ 2& 4\\ -1& -2\end{array}\right],\mathbf{b}=\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right]$

   2. $A=\left[\begin{array}{ccc}1& 1& 3\\ -1& 3& 1\\ 1& 2& 4\end{array}\right],\mathbf{b}=\left[\begin{array}{c}-2\\ 0\\ 8\end{array}\right]$

4. For each of the systems in Exercise 3, determine the projection p of b onto R(A) and verify that $\mathbf{b}-\mathbf{p}$ is orthogonal to each of the column vectors of A.

5. 1. Find the best least squares fit by a linear function to the data

      x	$-1$	0	1	2
      y	0	1	3	9

   2. Plot your linear function from part (a) ...