The question asks about the least squares solution, focusing on the matrix $A$ where $\text{rank}(A) = \text{number of columns of } A$. This implies that $A$ has full column rank, meaning $A^T A$ is invertible.

In least squares problems, the solution to $A x = b$ in the over-determined case (more equations than unknowns) is given by $x = (A^T A)^{-1} A^T b$. This formula represents the unique least squares solution because $A^T A$ is invertible.

Correct Answer:

The second option: $A(A^T A)^{-1} A^T b \text{ is the unique least square solution to } Ax = b.$ is always true in this case.

Would you like a detailed explanation or have any specific questions? Here are five related questions to consider:

1. What does it mean for a matrix to have full column rank?
2. Why is $A^T A$ invertible when $\text{rank}(A) = \text{number of columns of } A$?
3. What is the difference between a least squares solution and an exact solution?
4. How is the least squares solution derived from normal equations?
5. What happens to the least squares solution if $A^T A$ is not invertible?

Tip: In least squares problems, the projection $A u = b$ implies that the solution minimizes the error $\| A x - b \|$.