Question

Let A be an m x n matrix and b be a vector in R. Which of the following is/are true?

O Any solution of ATAx = Ab is a least-squares solution of Ax = b.
O A least-squares solution of Ax = b is a vector x such that b-Ax| ≤ ||b-A|| for all x in R".
O If b is in the column space of A, then every solution of Ax = b is a least-squares solution.
O The general least-squares problem is to find an x that makes Ax as close as possible to b.
O A least-squares solution of Ax = b is a vector x that satisfies Ax = b, where b is the orthogonal projection of b onto Col A.

Answer 1

The first statement is true. Any solution of the equation ATAx = Ab is a least-squares solution of the equation Ax = b. The second statement is true. A least-squares solution of the equation Ax = b is a vector x such that ||b - Ax|| ≤ ||b - A|| for all x in R. The fourth statement is true. The general least-squares problem is to find an x that makes Ax as close as possible to b. The fifth statement is true. A least-squares solution of the equation Ax = b is a vector x that satisfies Ax = b, where b is the orthogonal projection of b onto the column space of A.

Step-by-step explanation:

The first statement is true.

Any solution of the equation ATAx = Ab is a least-squares solution of the equation Ax = b. This is because the equation ATAx = Ab represents the normal equations for the least-squares problem, and its solutions are also solutions to the original equation Ax = b.

The second statement is true.

A least-squares solution of the equation Ax = b is a vector x such that ||b - Ax|| ≤ ||b - A|| for all x in R. This means that the difference between the actual value b and the predicted value Ax is minimized.

The fourth statement is true.

The general least-squares problem is to find an x that makes Ax as close as possible to b. This means that the difference between the actual value b and the predicted value Ax is minimized, which is the definition of the least-squares problem.

The fifth statement is true.

A least-squares solution of the equation Ax = b is a vector x that satisfies Ax = b, where b is the orthogonal projection of b onto the column space of A.