1. **Problem Statement:** You are solving a least squares problem involving matrices $A$, $b$, and the normal equations $A^T A x = A^T b$ to find the vector $x = (x_1, x_2, x_3)$. 2. **Check the Distance Calculation:** You wrote $||w_1 - w_2|| = ||(7, -3) - (0, 2)|| = ||(7, -5)|| = \sqrt{7^2 + 5^2} = \sqrt{26}$. The subtraction is incorrect: $(7, -3) - (0, 2) = (7 - 0, -3 - 2) = (7, -5)$, so the norm is $\sqrt{7^2 + (-5)^2} = \sqrt{49 + 25} = \sqrt{74}$, not $\sqrt{26}$. 3. **Matrix $A$ and Vector $b$:** You defined $$ A = \begin{bmatrix} 3 & 1 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 2 \end{bmatrix}, \quad b = \begin{bmatrix} -3 \\ -3 \\ 8 \end{bmatrix} $$ 4. **Transpose of $A$:** $$ A^T = \begin{bmatrix} 3 & 1 & -1 \\ 1 & 2 & 0 \\ 0 & 1 & 2 \end{bmatrix} $$ 5. **Compute $A^T A$:** $$ A^T A = \begin{bmatrix} 3 & 1 & -1 \\ 1 & 2 & 0 \\ 0 & 1 & 2 \end{bmatrix} \times \begin{bmatrix} 3 & 1 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 11 & 5 & -1 \\ 5 & 9 & 2 \\ -1 & 2 & 5 \end{bmatrix} $$ 6. **Compute $A^T b$:** $$ A^T b = \begin{bmatrix} 3 & 1 & -1 \\ 1 & 2 & 0 \\ 0 & 1 & 2 \end{bmatrix} \times \begin{bmatrix} -3 \\ -3 \\ 8 \end{bmatrix} = \begin{bmatrix} -20 \\ -9 \\ 13 \end{bmatrix} $$ 7. **Solve the system:** $$ \begin{bmatrix} 11 & 5 & -1 \\ 5 & 9 & 2 \\ -1 & 2 & 5 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -20 \\ -9 \\ 13 \end{bmatrix} $$ Your solution values are $x_1 = \frac{2}{9} \approx 0.22$, $x_2 = -\frac{11}{3} \approx -3.6$, and $x_3 = \frac{37}{9} \approx 4.11$, which are consistent with the system. 8. **Summary of Issues:** - The main error is in the distance calculation: the vector difference and norm were computed incorrectly. - The matrix $A$ and vector $b$ are consistent with the least squares setup. - The matrix multiplications and solutions for $x$ appear correct. 9. **Recommendation:** - Correct the distance calculation to $||w_1 - w_2|| = \sqrt{7^2 + (-5)^2} = \sqrt{74}$. - Verify any symbolic matrices with $\varepsilon$ carefully, as they seem unrelated to the main least squares problem. This should clarify what was wrong in your work and confirm the correctness of the least squares solution.