Question

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.

18.

1. If \(b\) is in the column space of \(A\), then every solution of \(Ax = b\) is a least squares solution.
2. The least-squares solution of \(Ax = b\) is the point in the column space of \(A\) closest to \(b\).
3. A least-square solution of \(Ax = b\) is a list of weights that, when applied to the columns of \(A\), produces the orthogonal projection of \(b\) onto \({\rm{Col}}\,A\).
4. If \(\hat x\) is a least-squares solution of \(Ax = b\), then \(\hat x = {\left( {{A^T}A} \right)^{ - 1}}{A^T}b\).
5. The normal equations always provide a reliable method for computing least-squares solutions.
6. If \(A\) has a \(QR\) factorization, say \(A = QR\), then the best way to find the least-squares solution of \(Ax = b\) is to compute \(\hat x = {R^{ - 1}}{Q^T}b\).

Short answer

1. The given statement is True.
2. The given statement is False.
3. The given statement is True.
4. The given statement is False.
5. The given statement is False.
6. The given statement is False.

Step-by-step solution

Statement (a)

When \(b\) is in the column space of \(A\), then \(b = Ax\) for some \(x\) is the least-square solution.

Statement (b)

If \(x\) is the least-square solution, then \(Ax\) is the point in the column space of closest to \(b\).

Thus, the statement is false.

 Statement (c)

If \(b\) is the orthogonal projection of \(b\) onto \({\rm{Col}}\,A\), that is \(Ax = b\), then the system \(Ax = b\) is consistent.

By the definition of orthogonal decomposition, the projection \(\hat b\) has the property that \(b - \hat b\)that is orthogonal to \({\rm{Col}}\,A\).

Hence, the statement is true.

Statement (d)

The matrix \({A^T}A\) is invertible if and only if the columns of \(A\) are linearly independent.

Thus, the equation \(Ax = b\) has only one least-square solution \(x = {\left( {{A^T}A} \right)^{ - 1}}{A^T}b\).

Hence, the statement is false.

Statement (e)

In the procedure of normal equations for a least-square solution, small errors in the calculation of \({A^T}A\) can lead to relatively large errors in the solution \(x\).

If the columns of\(A\)are linearly independent, the least squares solution can often be computed more reliably through\(QR\)factorization of\(A\).

Hence, the statement is false.

Statement (f)

When an\(m \times n\)matrix has linearly independent columns and\(M\)is the\(QR\)factorization of\(A\), then\(Ax = b\)has a unique solution for each\(b\)in\({R^m}\).

Since \(R\) is an upper triangular matrix, so \(x\) can be computed from \(Rx = {Q^T}b\).

Hence, the statement is false.