Chapter 5.4, Problem 9E

Consider the linear system $A\stackrel{\to }{x}=\stackrel{\to }{b}$ , where $A=\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right]$ and $\stackrel{\to }{b}=\left[\begin{array}{l}10\\ 20\end{array}\right]$
a. Draw a sketch showing the following subsets of ${ℝ}^2$ :
• the kernelof A, and ${\left(\ker A\right)}^{\perp }$
•theimage of $A^T$
• the solution set S of the system $A\stackrel{\to }{x}=\stackrel{\to }{b}$
b. What relationship do you observe between ker(A) and $im\left(A^T\right)$ ? Explain.
c. What relationship do you observe between ker( A)and S? Explain.
d. Find the unique vector ${\stackrel{\to }{x}}_0$ in the intersection of S and ${\left(\ker A\right)}^{\perp }$ . Show ${\stackrel{\to }{x}}_0$ on your sketch.
e. What can you say about the length of ${\stackrel{\to }{x}}_0$ comparedwith the length of all other vectors in S?