Homework 1

Due Oct 9th at 11:59pm. Submit to CANVAS at https://canvas.uw.edu/courses/1917157. You should use computers when appriopriate. You are permitted to use AI, but please indicate how and where it is used.

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.

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

b) Solve \(\begin{bmatrix} 1 & 0 & 1 \\ -1 & 2 & 2 \\ 0 & 2 & 3 \end{bmatrix} x = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\)

c) Solve \(\begin{bmatrix} 1 & 0 & 1 \\ -1 & 2 & 2 \\ 0 & 2 & 3 \end{bmatrix} x = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}\)

Problem 2

Consider the matrix \(\begin{bmatrix} 1 & 2 \\ -1 & 0 \\ 3 & 4 \end{bmatrix}\)

a) Find its null space.

b) Find its row space.

Problem 3

Let \(A\) be some matrix. Recall that the column rank of \(A\) is the dimension of its column space, and the row rank of \(A\) is the dimension of its row space. Prove that the column rank equals to the row rank.

Problem 4

Find all the eigenvalues and eigenvectors of the following matrix

\[\begin{bmatrix} 3 & -1 & -1.5 \\ -1 & 5 & -2 \\ -1.5 & -2 & 4 \end{bmatrix}\]

Problem 5

In class, I said that power need to be defined as \(S=VI^*\), and things wouldn’t make sense if that \(*\) isn’t there. Why is that? What would go wrong if we define power as \(S=VI\)?

Problem 6

Consider the circuit below, find the current through \(Z\), and the power delivered to the load \(Z_L\).

Problem 7

Consider a setting of a generator supplying a load. Let the load be \(L\). The price of electricity is proportional to the load, that is, the price is \(\alpha L\) for some \(\alpha >0\). Now fix the load at 10 MW. We add a solar generator to the system. Because solar energy has zero marginal cost, we consider it as negative load. Let denote the amount of solar generated as \(S\), and the price is now \(\alpha (10- S)\). If you’re the owner of the solar generator, what value of \(S\) would you choose to optimize your profit?