Practice Final Exam Questions

1

The augmented matrix \([A \mid \mathbf{b}]\) for a linear system has been reduced to

\[ \left[\begin{array}{cccc|c} 1 & 0 & 2 & 0 & 3 \\ 0 & 1 & -1 & 0 & 1 \\ 0 & 0 & 0 & 1 & -2 \end{array}\right]. \]

  1. Find a basis for the null space of \(A\).

  2. Write the general solution of \(A\mathbf{x} = \mathbf{b}\) in the form particular solution plus null space.

  3. What is the rank of \(A\)? Is \(\mathbf{b}\) in the column space of \(A\)? Briefly justify.

2

A directed graph has vertices \(A\), \(B\), \(C\), \(D\) and edges: \(A \to B\), \(A \to C\), \(B \to C\), \(B \to D\), \(C \to D\). (So there are 5 edges.)

  1. Write the adjacency matrix \(A\) and the incidence matrix \(B\) (orientation: tail \(= +1\), head \(= -1\)). Use vertex order \(A, B, C, D\) and number the edges in the order listed above.

  2. A loop in the graph is a subset of edges that form a closed path (each vertex has the same number of edges in as out along the loop). Find a basis for the set of loops by finding a basis for the null space of \(B\), and draw the graph with the basis of loops indicated. (It’s fine if the loops are not directed loops, just closed paths.)

For your reference, here is the RREF of \(B\):

\(\displaystyle \left[\begin{matrix}1 & 0 & -1 & 0 & 1\\0 & 1 & 1 & 0 & -1\\0 & 0 & 0 & 1 & 1\\0 & 0 & 0 & 0 & 0\end{matrix}\right]\)

  1. This graph does not contain any directed loops – that is, it is impossible to start at a vertex and follow a sequence of directed edges to return to the same vertex. Show that this property follows from the fact that \(A^4=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\).

3

A linear system in unknowns \(x_1, x_2, x_3, x_4\) has general solution \((4 + x_2 - 3x_4,\, x_2,\, 2 - x_4,\, x_4)\) where \(x_2\) and \(x_4\) are free. Find a minimal reduced row echelon form (augmented matrix) for this system.

4

Find a least-squares solution to

\[ \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}. \]

Is it a genuine solution? Verify by computing the residual.

5

Show that if \(A^T A \mathbf{y} = \mathbf{0}\), then \(A\mathbf{y} = \mathbf{0}\). (Hint: consider \(\|A\mathbf{y}\|^2\).)

6

  1. For \(A = \begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix}\), find the eigenvalues and eigenvectors, and find \(P\), \(D\) such that \(A = P D P^{-1}\).

  2. Use (a) to write \(A^3\) in the form \(P D^3 P^{-1}\) (you may leave \(P^{-1}\) unevaluated).

  3. What is the spectral radius of \(A\)?

  4. Using (a), diagonalize the matrix \(A^2 - 5A + 6I\). What is the null space of \(A^2 - 5A + 6I\)? (You may use the fact that for a diagonalizable matrix, \(p(A) = P\, p(D)\, P^{-1}\) when \(p\) is a polynomial.)

7

A population is modeled with three life stages (e.g. juvenile, subadult, adult). The transition matrix is \[ A = \begin{bmatrix} 0 & 0 & 2 \\ 0.5 & 0 & 0 \\ 0 & 0.6 & 0 \end{bmatrix}. \]

  1. In words, what does this model say about how individuals move between stages and reproduce in a given time step?

  2. Starting with a population vector \(\mathbf{x}_0 = (0, 20, 50)^T\), compute the population after one and after two time steps (i.e. \(\mathbf{x}_1\) and \(\mathbf{x}_2\)).

  3. The dominant eigenvalue of \(A\) (largest in modulus) is \(\lambda_1 \approx 0.84\). What does that tell you about the long-term behavior of the population?

8

Convert the difference equation \(y_{k+2} - 2y_{k+1} + y_{k} = 1\) into the form \(\mathbf{x}_{k+1} = A \mathbf{x}_k\) for some state vector \(\mathbf{x}_k\) and matrix \(A\). Because the right-hand side of the difference equation is nonzero, use a state vector that includes a constant component (e.g. a third entry equal to 1).

9

Suppose \(A = U \Sigma V^T\) is the SVD of a real \(m \times n\) matrix with nonzero singular values \(\sigma_1 \ge \cdots \ge \sigma_r > 0\). Let \(\mathbf{u}_i\) and \(\mathbf{v}_i\) be the \(i\)th columns of \(U\) and \(V\).

  1. What is \(A \mathbf{v}_1\) in terms of \(\sigma_1\) and \(\mathbf{u}_1\)?

  2. What is \(A^T A \mathbf{v}_1\)?

  3. Give a formula for the least-squares solution(s) to \(A\mathbf{x} = \mathbf{b}\) in terms of \(U\), \(\Sigma\), \(V\), and \(\mathbf{b}\).

  4. Suppose \(r = n\) (full column rank). Describe the solution set of \(A\mathbf{x} = \mathbf{0}\). What does that imply about the least-squares solution to \(A\mathbf{x} = \mathbf{b}\)?

10

Let \(A = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 3 & 2 \end{bmatrix}\).

  1. Find the reduced row echelon form of \(A\) and its rank.

  2. Find a basis for the column space of \(A\).

  3. Determine which of the following vectors is in the column space of \(A\) and, if so, express it as a linear combination of the columns of \(A\): \(\mathbf{b}_1 = [0, 0, 1]^T\), \(\mathbf{b}_2 = [1, 2, 3]^T\).

11

A bandpass filter attenuates both the lowest frequencies (near DC, \(k=0\)) and the highest frequencies (near Nyquist, \(k = N/2\) when \(N\) is even), and passes a band of frequencies in the middle. So the filter’s gain is (or is close to) zero at \(k=0\) and at \(k=N/2\), and is nonzero for some \(k\) strictly between \(0\) and \(N/2\).

A signal is sampled every \(T_s = 1/6\) s (sampling rate 6 Hz) and we take \(N = 6\) samples (1 s of data). There are \(N = 6\) frequency indices \(k = 0, 1, \ldots, 5\), with corresponding frequencies \(f_k\) (in Hz): \(0\), \(1\), \(2\), \(3\) (Nyquist), \(-2\), \(-1\).

  1. You would like to design a bandpass filter that passes signals with frequencies between 0.5 Hz and 2.5 Hz. Write down a simple example of the six gains \(H[0], \ldots, H[5]\) for such a filter.

(For the negative frequencies, use the same filter value H[k] as for the corresponding positive frequency.)

  1. Suppose your signal is \(x[n] = (1, 0, 1, 1, 1, 0)\) with DFT coefficients \(X_0, \ldots, X_5\) equal to \(4, -1, 1, 2, 1, -1\) respectively. Using the transformation from lecture (below), write down the \(n\)-th entry of the filtered signal (that is, \(y[n]\)) as a sum of terms involving complex exponentials. You don’t need to do any trigonometric simplifications. \[ \mathcal{F}_h(\mathbf{x}) = \frac{1}{N} \sum_{k=0}^{N-1} X_k H(\zeta_k) \mathbf{f}_k, \]

where \(\zeta_k = 2\pi k/N\) and \(H(\zeta_k)\) is the filter gain at frequency index \(k\) (the values \(H[0], \ldots, H[5]\) from part (a)).

  1. The filter can also be specified by its time-domain coefficients \(h[0], h[1], \ldots, h[N-1]\), with the usual DFT relation \(H[k] = \sum_{n=0}^{N-1} h[n] e^{-i 2\pi k n / N}\). How would you recover the coefficients \(h[n]\) from the gains \(H[k]\)? (State the formula or the name of the operation.)