Homework / Problem Set

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• Starter files
• main.tex1.7 KB

  \documentclass[11pt]{article}
  
  \usepackage[margin=1in]{geometry}
  \usepackage{amsmath,amssymb,amsthm}
  \usepackage{fancyhdr}
  \usepackage{enumitem}
  
  % ---- Fill these in ----
  \newcommand{\course}{MATH 201: Linear Algebra}
  \newcommand{\hwnumber}{3}
  \newcommand{\yourname}{Your Name}
  \newcommand{\duedate}{September 12, 2026}
  
  \pagestyle{fancy}
  \fancyhf{}
  \lhead{\course}
  \chead{Problem Set \hwnumber}
  \rhead{\yourname\ \textbullet\ Due \duedate}
  \cfoot{\thepage}
  
  \theoremstyle{definition}
  \newtheorem{problem}{Problem}
  \newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
  
  \begin{document}
  
  \begin{problem}
  Let $A \in \mathbb{R}^{n \times n}$ be symmetric. Show that all
  eigenvalues of $A$ are real.
  \end{problem}
  
  \begin{solution}
  Let $Av = \lambda v$ with $v \neq 0$. Then
  \[
    \lambda \, \overline{v}^{\top} v
    = \overline{v}^{\top} A v
    = (A\overline{v})^{\top} v
    = \overline{\lambda} \, \overline{v}^{\top} v,
  \]
  and since $\overline{v}^{\top} v > 0$ we conclude
  $\lambda = \overline{\lambda}$, so $\lambda \in \mathbb{R}$.
  \end{solution}
  
  \begin{problem}
  Compute the determinant of
  $\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}$
  and decide whether the matrix is invertible.
  \end{problem}
  
  \begin{solution}
  $\det = 2 \cdot 3 - 1 \cdot 1 = 5 \neq 0$, so the matrix is invertible.
  \end{solution}
  
  \begin{problem}
  Prove or disprove: if $A$ and $B$ are invertible $n \times n$ matrices,
  then $A + B$ is invertible.
  \end{problem}
  
  \begin{solution}
  False. Take $B = -A$ with $A$ invertible; then $A + B = 0$ is not
  invertible.
  \end{solution}
  
  % Add more problems by copying a problem/solution pair, or ask the agent
  % in the chat alongside this document to typeset your work for you.
  
  \end{document}
  

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