To find the least squares solution for the system $A \mathbf{x} = \mathbf{b}$, where

$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} \quad \text{and} \quad \mathbf{b} = \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix},$

we will solve it in two ways:

(a) Without Calculating the Left Inverse

The least squares solution $\mathbf{x}_{\text{ls}}$ can be found using the formula: $\mathbf{x}_{\text{ls}} = (A^T A)^{-1} A^T \mathbf{b}.$

1. Calculate $A^T$: $A^T = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}.$

2. Calculate $A^T A$: $A^T A = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} = \begin{bmatrix} 35 & 44 \\ 44 & 56 \end{bmatrix}.$

3. Invert $A^T A$: $(A^T A)^{-1} = \frac{1}{(35)(56) - (44)^2} \begin{bmatrix} 56 & -44 \\ -44 & 35 \end{bmatrix} = \begin{bmatrix} 1.4 & -1.1 \\ -1.1 & 0.875 \end{bmatrix}.$

4. Calculate $A^T \mathbf{b}$: $A^T \mathbf{b} = \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix} \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix} = \begin{bmatrix} 76 \\ 100 \end{bmatrix}.$

5. Calculate $\mathbf{x}_{\text{ls}}$: $\mathbf{x}_{\text{ls}} = (A^T A)^{-1} A^T \mathbf{b} = \begin{bmatrix} 1.4 & -1.1 \\ -1.1 & 0.875 \end{bmatrix} \begin{bmatrix} 76 \\ 100 \end{bmatrix} = \begin{bmatrix} -3 \\ 5 \end{bmatrix}.$

Thus, the least squares solution is: $\mathbf{x}_{\text{ls}} = \begin{bmatrix} -3 \\ 5 \end{bmatrix}.$

(b) By Calculating the Left Inverse

Since $A$ is not a square matrix, it doesn’t have an ordinary inverse. However, the least squares solution method used in part (a) already effectively involves calculating the pseudoinverse of $A$, so this method provides the solution without requiring any additional calculations.

If you have further questions or want a breakdown of any step, please let me know!

Related Questions:

1. How would we calculate the least squares solution if $A$ were square and invertible?
2. What is the geometric interpretation of a least squares solution in terms of projections?
3. Can the least squares method be used to solve inconsistent systems? Why?
4. What are the conditions under which $(A^T A)$ is invertible?
5. How would the solution change if we modified the values in vector $\mathbf{b}$?

Tip:

When working with least squares, remember that it finds the "best approximate" solution by minimizing the distance from $A \mathbf{x}$ to $\mathbf{b}$, which is useful for inconsistent systems.