Homework 12

1. Least Squares
2. Linear Models
- 2.1. (2 points)
- 2.2. (2 points)
3. Quadratic Forms
4. SVD (Extra Credit)
- 4.1. (1 point)
- 4.2. (1 point)

The following assignment is due Tuesday 12/10 by 11:59 PM. You should submit all your written solutions to Gradescope as a single pdf. Follow the instructions in the programming problem for code solutions.

Your solutions must be exceptionally neat and the final answer in your solution to each problem must be abundantly clear, e.g., surrounded in a very visible box. The graders have license to dock points for illegible or unclear solutions.
For the written part, choose the correct pages corresponding to each problem in Gradescope. Note that Gradescope registers your submission as soon as you submit it, so you don't need to rush to choose the pages. You will receive no credit if you do not choose the correct pages, no exceptions.

1. Least Squares

\begin{array}{r} A = [\begin{array}{c} 1 & 2 \\ 1 & - 1 \\ 1 & - 1 \end{array}] b = [\begin{array}{c} 4 \\ 2 \\ 0 \end{array}] \end{array}

For this problem, you must do all calculations by hand and show your work.

1.1. (2 points)

Determine the normal equations of $A x = b$ .

1.2. (2 points)

Determine a least-squares solution of $A x = b$ .

1.3. (1 points)

Is the least-squares solution you determined in the previous part unique? Justify your answer.

2. Linear Models

We will use the following dataset for this problem.

\begin{array}{r} {(- 2, 0), (- 1, 5), (0, 13), (1, 9), (2, 5), (3, 0)} \end{array}

2.1. (2 points)

Set up a linear system whose least-squares solution determines the parameters for the least-squares fit model of the form $f (x) = β_{1} \cos x + β_{2} \sin x + β_{3} x + β_{4}$ (in other words, determine the design matrix and the vector of labels for the given data and model). Round the entries of the design matrix to the nearest hundredth.

2.2. (2 points)

Determine the least-squares fit model for the given data. Round the parameters to the nearest hundredth.

3. Quadratic Forms

Consider the following quadratic form.

\begin{array}{r} 3 x_{1}^{2} + 3 x_{2}^{2} + 5 x_{3}^{2} + 6 x_{1} x_{2} + 2 x_{1} x_{3} + 2 x_{2} x_{3} \end{array}

3.1. (2 points)

Determine the underlying matrix for $A$ .

3.2. (2 points)

Determine $\max_{‖ x ‖ = 1} x^{T} A x$ , that is the maximum value of $x^{T} A x$ . You may use a computer for this part, but you must describe your process and justify your answer.

3.3. (2 points)

Determine ${argmax}_{‖ x ‖ = 1} x^{T} A x$ , that is unit vector which attains the maximum value. For this part, you must do calculations by hand and show your work.

4. SVD (Extra Credit)

\begin{array}{r} A = [\begin{array}{c} 3 & 2 \\ 2 & 3 \\ 2 & - 2 \end{array}] \end{array}

4.1. (1 point)

Determine a singular value decomposition of $A$ . For this part, you must do calculations by hand and show your work.

4.2. (1 point)

Determine a singular value decomposition of $A^{T}$ . (Hint. Don't do all the work again)