Question

Let A be an m x n matrix with linearly independent columns, and let b be a vector in Rm. Find x such that every solution to the equation Ax = b is a least squares solution.

a) x = (ATA)−¹ATь
b) x = A(ATA)-¹b
c) x = (AAT)-¹Ab
d) x = (ATA) Ab

Answer 1

The correct expression to find the vector x so that every solution to Ax = b is a least squares solution is x = (A^TA)−¹ATb, which stems from the least squares method used to minimize the sum of the squares of residuals.

Step-by-step explanation:

The subject of this question is to find the vector x such that every solution to the equation Ax = b is a least squares solution, where A is an m x n matrix with linearly independent columns and b is a vector in Rm. The correct expression for x is x = (ATA)−¹ATb. This form arises from the least squares method, which minimizes the sum of the squares of the residuals (the differences between observed and predicted values).

Here's the rationale: when the equation Ax = b has no exact solutions, the least squares solution is the x that minimizes the error measured by the norm of the vector Ax - b. The solution can be found by solving the normal equation ATAx = ATb. Since the columns of A are linearly independent, ATA is invertible, and the least squares solution x is given by x = (ATA)−¹ATb.