Question

A least-squares solution of Ax=b is a vector x^ such that ||b-Ax|| is minimum for all x in Rn. True or False?

Answer 1

This statament A least-squares solution of Ax=b is a vector x^ such that ||b-Ax|| is minimum for all x in Rn is true because The least-squares solution of Ax=b minimizes the Euclidean norm of the residual vector ||b-Ax|| for all x in
`R^n`, providing the closest approximation in overdetermined or inconsistent systems.

Step-by-step explanation:

In the least-squares solution of Ax=b, the goal is to minimize the Euclidean norm of the residual vector ||b-Ax||. This solution, denoted as x^, is found by minimizing the sum of squared differences between the actual and predicted values of b. Mathematically, this is expressed as minimizing
`||b-Ax||^2`.

To find the least-squares solution, we differentiate the expression with respect to x, set the result equal to zero, and solve for x. The resulting x^ satisfies the condition that the gradient of the residual norm is zero. This condition ensures that the vector x minimizes the sum of squared differences.

The least-squares solution is particularly useful when the system of equations is overdetermined or inconsistent, meaning there are more equations than unknowns or no exact solution exists. In such cases, finding an exact solution may be impossible, but the least-squares solution provides the closest approximation that minimizes the overall error. Therefore, the statement is true – the least-squares solution of Ax=b is a vector x^ that minimizes ||b-Ax|| for all x in
`R^n`.