Homework 9

1. Linear Equations and Column Spaces
- 1.1. (4 points)
- 1.2. (4 points)
2. Subspaces
- 2.1. (4 points)
- 2.2. (4 points)
3. Coordinate systems
- 3.1. (2 points)
- 3.2. (3 points)
4. Eigenvalues
5. Eigenbases

The following assignment is due Thursday 11/14 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. Linear Equations and Column Spaces

For each of the following parts you must show your work. You may use a computer but you must write down any matrix you reduce along with its reduced echelon form.

1.1. (4 points)

Consider the matrix

\begin{align*} A = \begin{bmatrix} 1 & -4 & -10 \\ -4 & 17 & 42 \\ 1 & -2 & -6 \end{bmatrix} \end{align*}

Determine the linear equation whose solution set is \(\mathsf{Col}(A)\).

1.2. (4 points)

Let \(A\) be a \(4 \times 2024\) matrix where \(\mathsf{rank}(A) = 3\). Further suppose that the LU-decomposition of \(A\) has

\begin{align*} L = \begin{bmatrix} 1 & 0 & 0 & 0 \\ -2 & 1 & 0 & 0 \\ 7 & 5 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{bmatrix} \end{align*}

The matrix \(U\) is unknown. Determine the linear equation whose solution set is \(\mathsf{Col}(A)\).

2. Subspaces

Consider the following two matrices.

\begin{align*} A = \begin{bmatrix} 6 & -6 & -1 & 17 \\ 9 & -2 & 2 & 36 \\ -10 & -13 & 8 & -45 \\ -9 & 4 & -1 & -33 \\ 2 & 4 & 2 & 14 \end{bmatrix} \qquad B = \begin{bmatrix} -8 & 5 & -16 & 1 \\ 2 & 11 & 4 & 16 \\ 3 & -2 & 6 & 3 \\ 2 & -10 & 4 & 0 \\ 8 & 4 & 16 & 9 \end{bmatrix} \end{align*}

Also consider the following set of vectors.

\begin{align*} H = \{ \mathbf u + \mathbf v : \mathbf u \in \mathsf{Col}(A) \text{ and } \mathbf v \in \mathsf{Col}(B)\} \end{align*}

That is, \(H\) consists of all sums of pairs of vectors, one from \(\mathsf{Col}(A)\) and one from \(\mathsf{Col}(B)\).

2.1. (4 points)

Show that \(H\) is a subspace of \(\mathbb R^5\).

2.2. (4 points)

Determine the dimension of \(H\). Justify your answer. You may use a computer but you must write down any matrix you reduce along with its reduced echelon form.

3. Coordinate systems

In each part below, determine the coordinate vector \([\mathbf v]_{\mathcal{B}}\) for the given vector \(\mathbf v\) and basis \(\mathcal B\). You may use a computer but you must write down any matrix you reduce along with its reduced echelon form.

3.1. (2 points)

\begin{align*} \mathbf v = \begin{bmatrix} -17 \\ 15 \end{bmatrix} \qquad \mathcal B = \left\{ \begin{bmatrix} 1 \\ -5 \end{bmatrix}, \begin{bmatrix} 3 \\ -1 \end{bmatrix} \right\} \end{align*}

3.2. (3 points)

\begin{align*} \mathbf v = \begin{bmatrix} 5 \\ 29 \\ -80 \\ 42 \end{bmatrix} \qquad \mathcal B = \left\{ \begin{bmatrix} -11 \\ -19 \\ 17 \\ 9 \end{bmatrix}, \begin{bmatrix} 9 \\ 17 \\ -20 \\ -2 \end{bmatrix}, \begin{bmatrix} -3 \\ 1 \\ -18 \\ 18 \end{bmatrix} \right\} \end{align*}

4. Eigenvalues

For each matrix \(A\) and real number \(\lambda\), determine a basis for the eigenspace of \(A\) associated with the eigenvalue \(\lambda\). If \(\lambda\) is not an eigenvalue of \(A\), then write \(\lambda\) is not an eigenvalue of \(A\).¹

4.1. (3 points)

\begin{align*} A = \begin{bmatrix} 4 & -42 \\ 1 & -9 \end{bmatrix} \qquad \lambda = -3 \end{align*}

4.2. (3 points)

\begin{align*} A = \begin{bmatrix} 3 & -8 & 8 \\ 8 & -13 & 8 \\ 2 & -2 & -3 \end{bmatrix} \qquad \lambda = -5 \end{align*}

4.3. (3 points)

\begin{align*} A = \begin{bmatrix} 3 & -8 & 8 \\ 8 & -13 & 8 \\ 2 & -2 & -3 \end{bmatrix} \qquad \lambda = 4 \end{align*}

5. Eigenbases

5.1. (3 points)

Suppose that \(A\) is a matrix such that

\begin{align*} A \begin{bmatrix} 1 \\ 1 \\ -3 \end{bmatrix} = \begin{bmatrix} -1 \\ -1 \\ 3 \end{bmatrix} \qquad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \qquad A \begin{bmatrix} 2 \\ -2 \\ -9 \end{bmatrix} = \begin{bmatrix} 4 \\ -4 \\ -18 \end{bmatrix} \qquad \end{align*}

Determine the vector

\begin{align*} A^5 \begin{bmatrix} 3 \\ 0 \\ -11 \end{bmatrix} \end{align*}

You may use a computer, but you must show your work and justify your answer. Your solution cannot include determining the matrix \(A\).

5.2. (4 points)

Let \(A\) be the matrix from the previous part. Consider the basis

\begin{align*} \mathcal B = \left\{ \begin{bmatrix} 1 \\ 1 \\ -3 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ -2 \\ -9 \end{bmatrix} \right\} \end{align*}

Determine the matrix \(C\) such that \(C[\mathbf v]_{\mathcal B} = A \mathbf v\) for all \(\mathbf v \in \mathbb{R}^3\).

5.3. (3 points)