1. In Exercises 17 and 18, \(A\) is an \(m \times n\) matrix and \(b\) is in \({\mathbb{R}^m}\). Mark each statement True or False. Justify each answer.

17.

1. The general least-square problem is to find an \(x\) that makes \(Ax\) as close as possible to \(b\).
2. A least-square solution of \(Ax = b\) is a vector \(\hat x\) that satisfies \(A\hat x = \hat b\), where \(\hat b\) is the orthogonal projection of \(b\)onto \({\rm{Col }}A\).
3. A least-square solution of \(Ax = b\) is a vector \(\hat x\) such that \(\left\| {b - Ax} \right\| \le \left\| {b - A\hat x} \right\|\) for all \(x\) in \({\mathbb{R}^n}\).
4. Any solution of \({A^T}Ax = {A^T}b\) is a least-squares solution of \(Ax = b\).
5. If the columns of \(A\) are linearly independent, then the equation \(Ax = b\) has exactly one least-squares solution.

(a)

The general least-square problem is to find \(\left\| {b - Ax} \right\|\) which is equal to the root of a sum of squares. This is equal to the distance between \(b\) and \(Ax\).

Therefore, the statement is true.

(b)

If \(b\) is the orthogonal projection of \(b\) onto \({\rm{Col }}A\), then \[Ax = b\]. Also,the orthogonal projection theorem states that \(b - \hat b\) is orthogonal to \({\rm{Col}}\,A\). So,

\(\begin{aligned}{}{A^T}\left( {b - A\hat x} \right) = 0\\{A^T}b - {A^T}A\hat x = 0\\{A^T}A\hat x = {A^T}b\end{aligned}\)

Hence, the statement is true.

(c)

The definition of least-square solution states that a least-square solution of \(Ax = b\) is a vector \(x\) such that \(\left\| {b - Ax} \right\| \ge \left\| {b - A\hat x} \right\|\).

Thus, the statement is false.

(d)

The nonempty set of solutions to the normal equations \({A^T}Ax = {A^T}b\) corresponds to the set of least-square solutions of \(Ax = b\).

Hence, the statement is true.

(e)

If the columns of \(A\) are linearly independent, the equation \(Ax = b\) has only one least square solution, which is provided by \(\hat x = {\left( {{A^T}A} \right)^{ - 1}}{A^T}b\), according to the Theorem 14.

Hence, the statement is true.