Chapter 1.6, Problem 48E

Let $A$ be the $\left(2\times 2\right)$ matrix

$A=\left[\begin{array}{cc}1& 2\\ 3& 6\end{array}\right]$

Choose some vector $b$ in $R^2$ such that the equation $Ax=b$ is inconsistent. Verify that the associated equation $A^{T}Ax=A^{T}b$ is consistent for your choice of $b$. Let $x^{\ast }$ be a solution to $A^{T}Ax=A^{T}b$, and select some vectors $x$ at random from $R^2$. Verify that $\Vert A{x}^{\ast }-b\Vert \le \Vert Ax-b\Vert$ for any of these random choices of $x$. (In Chapter 3, we will show that $A^{T}Ax=A^{T}b$ is always consistent for any $\left(m\times n\right)$ matrix $A$ regardless of whether $Ax=b$ is consistent or not. We also show that any solution $x^{\ast }$ of $A^{T}Ax=A^{T}b$ satisfies $\Vert A{x}^{\ast }-b\Vert \le \Vert Ax-b\Vert$ for all $x$ in $R^n$; that is, such a vector $x^{\ast }$ minimizes the length of the residual vector $r=Ax-b$.)