Homework 1

Due Oct 3th at 11:59pm. Submit to CANVAS at https://canvas.uw.edu/courses/1828831/assignments/10721423

Problem 1

We will use matrices extensively in this class, often in the context of solving linear system of equations. For the following equations, find all the solutions, or explain why there is no solution. You may use a computer.

a) Solve \(\begin{bmatrix} 1 & 2 & 3 \\ 4 & 0 & 5 \\ 6 & 11 & 2 \end{bmatrix} x =\begin{bmatrix} 1 \\2 \\ 3 \end{bmatrix}\)

b) Solve \(\begin{bmatrix} 1 & 2 \\ -0.5 & -1 \end{bmatrix} x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\)

c) Solve \(\begin{bmatrix} 2 & 0 & 1 \\ -1 & 1 & 2 \end{bmatrix} x = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\)

Problem 2

Some questions about matrices.

a) Let \(\mathbf{A}\) be a \(n\) by \(n\) symmetric real matrix. Show the eigenvalues of \(\mathbf{A}\) are real.

b) Find a 2 by 2 matrix with just one eigenvalue. Note, matrices with repeated eigenvalues do not count. For example, the identity matrix has 2 eigenvalues both equal to 1, but it’s not what this question is asking for. We are looking for a matrix with just 1 eigenvalue.

c) Let \(\mathbf{A}\) and \(\mathbf{B}\) be two matrices where \(\mathbf{AB}\) and \(\mathbf{BA}\) both make sense. Let \(\mathrm{Tr}\) be the trace operator. Show that \(\mathrm{Tr} (\mathbf{AB})=\mathrm{Tr} (\mathbf{BA})\).

Problem 3

Complex numbers. Note that \(z^*\) denote the complex conjugate of \(z\).

a) Let \(z_1\) and \(z_2\) be two complex numbers, both with magnitude equal to 2. What is the largest possible value of \(\mathrm{Re} (z_1 \cdot z_2)\)?

b) A subset \(U\) of \(\mathbb{C}^n\) is called a complex subspace of \(\mathbb{C}^n\) if it contains \(0\) and if, given \(\mathbf{v}\) and \(\mathbf{u}\) in \(U\), \(\mathbf{v}+\mathbf{u}\) and \(z \mathbf{v}\) lie in \(U\) (for any complex scalar \(z\)). Determine whether \(U\) is a complex subspace of \(\mathbb{C}^3\) in each of the case below.

i. \(U=\{(z,z^*,0) \vert z \in \mathbb{C}\}\)

ii. \(U=\mathbb{R}^3\)

iii. \(U=\{(v+w, v-3w, v) \vert v,w \in \mathbb{C}\}\)

Problem 4

Basic circuits. The values of the elements are \(R_1=1 \Omega\), \(R_2=2 \Omega\), \(Z_C=-j0.5 \Omega\), \(Z_L = j 3 \Omega\).

a) Find the active power and reactive power produced by the source.

b) Suppose we want the source to only deliver active power. Add to element to the circuit to achieve it (there are many ways to do this).

Problem 5

Given a network with \(n\) buses. We say the network is connected if there is a path from any bus to any other bus.

a) What is the smallest number of edges a connected network can have?

b) What about the largest number of edges?

c) For a practical grid, would it have a lot or a few edges?