The first portfolio. As the semester progresses I'll be asking more specific and additional questions regarding what to include. For this one though, the idea is to start to get the hang of the portfolio system and, of course, to demonstrate that you've understand the big ideas in Chapter 1 of Hefferon. I encourage you to try to typeset at least one question in LaTeX and show that you can use Sage to solve a system of linear equations (or something else).

As usual, show me that you've mastered the material. Your portfolio should be self-contained and have appropriate references. Along the way, also:

- typeset at least problem's worth of material in LaTeX,
- use Sage, if you didn't in Portfolio 1,
- include a problem that you /can't/ solve, discussing what you tried and what you managed to accomplish (alternatively, you can put in some completely solved hard problems!)

Some examples of possible harder problems:

- Either the crystal or voting Topics at the end of Chapter 2 of Hefferon,
- The equiangular line problem from Week 1,
- A small liberal arts college with population \( n \) has \( m \) committees. Any two committees must have an even (possibly 0) number of members in common. Show that there are at least as many people as committees.
- A graph is a network of nodes joined by edges. Given a graph on \(n\) nodes create its \( n \times n \) adjacency matrix by putting a 1 in position \( (i,j) \) if there is an edge between node \( i \) and node \( j \) and put a 0 there otherwise. How can you calculate the number of different paths from one node to another?

Try to lean towards more conceptual questions: the next portfolio will be focused on getting the mechanical technique questions as automated as possible. Similarly, push on using LaTeX as much as possible here because there will be little-to-none in the next one. Also try your hand at expository writing, as opposed to only directly answering questions.

Make sure you can do the following, preferably to the point of automation:

- Put a matrix in row echelon or reduced row echelon form, using the Gauss-Jordan method.
- Find the length of a vector.
- Calculate the dot product of two vectors.
- Check whether a set of vectors is linearly independent.
- Check whether a set of vectors spans a space.
- Multiply matrices.
- Find the inverse of a matrix.
- Find the projection of one vector onto another.
- Apply the Gram-Schmidt process to find an orthonormal basis from a given one.
- Calculate the determinant of a matrix.

For some (all?) of these, you should reach the point where it feels tedious and arithmetic errors are the most common source of errors. For these, you might want to work out how to ask Sage to do it for you.

LATE ADDITIONS (don't worry if you've already put the time in and don't get to these):

- Arithmetic of complex numbers.
- Finding eigenvalues and eigenvectors.

The core of the material here is complex numbers and eigenvalues and diagonalisation. That is, the first two large subsections of Chapter 5 of Hefferon. However, I'd like you to do more in addition to this. This could be to revisit an earlier topic (or Hefferon end-of-chapter topic) that you didn't quite fully get or we skipped past. Or you could press on with Chapter 5 (which is good stuff). Or something else.

- Details in class.

You don't need to do anything else; this is here so that it'll show up on your grades page.