Let

A =













1 2

3 4

5 6













and b =













7

8

9













.

Find the least squares solution of the system

(a) without calculating left inverse

(b) by calculating the left inverse

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.

Let A = [[1, 2], [3, 4], [5, 6]] and b = [7, 8, 9]. Find the least squares solution of the system (a) without calculating left inverse (b) by calculating the left inverse.

Learn how to find the least squares solution for the system A x = b, given matrix A and vector b. This problem explores two methods: (a) using the standard least squares formula without calculating the left inverse, and (b) understanding the role of the left inverse for consistent solutions. Suitable for college-level linear algebra courses.

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