Homework 1

Due: Sep 6th, 2021

Problems

  1. Read Self-Explanation Training for Mathematics Students.

    If this is the first time you’ve read this document, then write a self-explanation of Practice Proof 1.

    If you’ve already written a self-explanation of Practice Proof 1, then write a self-explanation of Practice Proof 2. (For example, if you have taken a previous class from me where I already assigned this reading.)

  2. Section 1.1, Exercises 1-6: Attempt all. Solve as many as you can. Turn in the best/most interesting textbook exercise you solve.

  3. Exercise 1.2.6

  • Visit WeBWork. Complete the “Chapter 0” assignment within WebWork. (This will be graded within WebWork. You don’t have to turn in anything to Gradescope for this.)

Next is a list of several more problems. Please attempt all of them and solve as many as you can. Turn in the best/most interesting two (2) problems you solved. Label very clearly which problems you are turning in. Please do not turn in extra problems for grading. (If you want to discuss additional problems or get feedback, send me an email.)

  1. Any one of the textbook exercises from Section 1.2, Exercises 1–4.

    These involve sketching. I strongly suggest hand-sketching your answers, scanning them, and including the scans using \includegraphics. If you want to try graphics software such as TikZ you are welcome to try, but please be aware that this is significantly more challenging. (Part of the challenge arises because it uses some algebraic geometry from later in the course. If you’re interested, this could be an opportunity to learn about algebraic geometry for computer graphics!)

    1. Verify the identities \[ (x_1 + x_2)^2 - (x_1 - x_2)^2 = 2! 2^1 x_1 x_2 \] and \[ (x_1 + x_2 + x_3)^3 - (x_1 + x_2 - x_3)^3 - (x_1 - x_2 + x_3)^3 + (x_1 - x_2 - x_3)^3 = 3! 2^2 x_1 x_2 x_3 . \]
    2. Formulate a conjecture that extends this.
    3. Prove your conjecture.

  3. Consider the equation \[ (ax+by)^2 = xy. \] Expanding gives \[ a^2 x^2 + 2abxy + b^2y^2 = 0x^2 + 1xy + 0y^2. \] For these polynomials to be equal, we have to satisfy the system of equations

    \[\left\{ \begin{array} {rl} a^2 &= 0 \\\\ 2ab &= 1 \\\\ b^2 &= 0 \\\\ \end{array} \right.\]

    1. Does this system of equations have a solution? Explain.

    2. Consider the equation \[ (ax+by)^2 + (cx+dy)^2 = xy. \] Write the resulting system of equations for \(a,b,c,d\). Does it have a solution?

Instructions

Use LaTeX to create a PDF. Upload your PDF to Gradescope. If you don’t have LaTeX on your computer, you can use Overleaf. Don’t submit the LaTeX source, just the PDF. Email your instructor (that’s me) if you have questions or need help.

Please include your name and the homework number (this is Homework 0) within the document. Some additional formatting instructions are in the syllabus. To summarize:

  • Use a new page (\newpage) for each problem.
  • State which question you are answering and the actual question. Then, start your answer in a new paragraph.
  • Use environments such as proof and theorem (via \begin{proof}...\end{proof}).
  • Use 12pt option \documentclass[12pt]{amsart} and linespacing \linespread{2.4}.

Please find and use the LaTeX template for homework.

You are encouraged to work together on the homework!

Additional reading