我的 Quarto PDF 无法渲染！
当我在 RStudio 中按下“渲染”时，终端中输出的错误如下：
pandoc
to: pdf
output-file: LearningLaTeX.pdf
standalone: true
pdf-engine: latexmk
variables:
graphics: true
tables: true
default-image-extension: pdf
toc: true
metadata
documentclass: scrartcl
classoption:
- DIV=11
- numbers=noendperiod
papersize: letter
header-includes:
- '\KOMAoption{captions}{tableheading}'
block-headings: true
editor: visual
title: Learning LaTeX
author: Anish Goyal
date: 'January 18, 2023'
subtitle: A document containing a collection of mathematical expressions in LaTeX and a 'funny' meme about LaTeX for the Applications of Linear Algebra class at GSMST.
latexmk: security risk: running with elevated privileges
pdflatex: security risk: running with elevated privileges
Latexmk: If appropriate, the -f option can be used to get latexmk
to try to force complete processing.
Reverting Windows console CPs to (in,out) = (437,65001)
Error producing PDF.
! Missing \endgroup inserted.
<inserted text>
\endgroup
l.289 \[\begin{eqnarray}
这是 Quarto 源 (.qmd)：
---
title: "Learning LaTeX"
author: Anish Goyal
date: January 18, 2023
subtitle: "A document containing a collection of mathematical expressions in LaTeX and a 'funny' meme about LaTeX for the Applications of Linear Algebra class at GSMST."
format:
pdf:
pdf-engine: latexmk
latex-auto-install: true
code-overflow: wrap
toc: true
---
## Mathematical expressions
### Expression 1
$$
\begin{cases}
w + x +y +z=6 \\
w \hspace{90px} +y+z =4 \\
w \hspace{90px}+y \hspace{90px} =2 \\
\end{cases}
$$
$$
\Huge
\begin{cases}
w + x +y +z=6 \\
w \hspace{90px} +y+z =4 \\
w \hspace{90px}+y \hspace{90px} =2 \\
\end{cases}
$$
### Expression 2
$$
k\neq-1, 1 \\ {\frac{1}{k+1}},{\frac{1}{k+1}}
$$
$$\Huge
k\neq-1, 1 \\ {\frac{1}{k+1}},{\frac{1}{k+1}}
$$
### Expression 3
$$
\operatorname{tr}A:=\sum_{j=1}^{n}a_{jj}
$$
$$
\Huge
\operatorname{tr}A:=\sum_{j=1}^{n}a_{jj}
$$
### Expression 4
$$
A(\theta)={\left[
\begin{matrix}
\cos{\theta} & -\sin{\theta} \\
\sin{\theta} & \cos{\theta}
\end{matrix}\right]}
$$
$$ \Huge
A(\theta)={\left[
\begin{matrix}
\cos{\theta} & -\sin{\theta} \\
\sin{\theta} & \cos{\theta}
\end{matrix}\right]}
$$
### Expression 5
$$
A = \left [ \begin{matrix}
0 & a & a^2 & a^3 & a^4 \\
0 & 0 & a & a^2 & a^3 \\
0 & 0 & 0 & a & a^2 \\
0 & 0 & 0 & 0 & a \\
0 & 0 & 0 & 0 & 0
\end{matrix} \right ]
$$
$$ \Huge
A = \left [ \begin{matrix}
0 & a & a^2 & a^3 & a^4 \\
0 & 0 & a & a^2 & a^3 \\
0 & 0 & 0 & a & a^2 \\
0 & 0 & 0 & 0 & a \\
0 & 0 & 0 & 0 & 0
\end{matrix} \right ]
$$
### Expression 6
$$
\bf{3.1.6. Proposition.} \ Let \ n \in \mathbb{N} \ and \ \bf{M}_{n \times n} \ be \ the \ collection \ of \ all \ n \times n \ matrices. \ There \ is \ exactly \ one \\ function \\
\mathrm{det}: \bf{M}_{n \times n} \rightarrow \mathbb{R}\colon A\mapsto\operatorname{det}A
$$
$$\begin{eqnarray}
\bf{3.1.6. Proposition.} \ Let \ n \in \mathbb{N} \ and \ \bf{M}_{n \times n} \ be \ the \ collection \ of \ all \ n \times n \ matrices. \ There \ is \ exactly \ one \ function \\
\mathrm{det}: \bf{M}_{n \times n} \rightarrow \mathbb{R}\colon A\mapsto\operatorname{det}A
\end{eqnarray}$$
### Expression 7
$$
\bf{3.1.12. Proposition.} \ If \ A \in \mathbb{N} \ and \ 1 \le j \le n, \ then \\
\mathrm{det}\ A: \sum_{k=1}^{n}a_{jk}C_{jk}
$$
$$\Large \begin{eqnarray}
\bf{3.1.12. Proposition.} \ If \ A \in \mathbb{N} \ and \ 1 \le j \le n, \ then \\
\mathrm{det}\ A: \sum_{k=1}^{n}a_{jk}C_{jk}
\end{eqnarray}$$
### Expression 8
$$
\vert a d+b e+c f\vert\leq{\sqrt{a^{2}+b^{2}+c^{2}}}{\sqrt{d^{2}+e^{2}+f^{2}}}
$$
$$ \huge
\vert a d+b e+c f\vert\leq{\sqrt{a^{2}+b^{2}+c^{2}}}{\sqrt{d^{2}+e^{2}+f^{2}}}
$$
### Expression 9
$$
\text{Let $V$ be the set of real numbers $x$ such that $x >= 0$. Define an operation of "addition" by} \\
x \boxplus y = xy+1 \\
\text{for all $x, y \in V$. Define an operation of "scalar multiplication" by} \\
a \boxdot x = a^2x \\
\text{for all $a \in R$ and $x \in V$.}
$$
$$\begin{eqnarray}
\text{Let $V$ be the set of real numbers $x$ such that $x >= 0$. Define an operation of "addition" by} \\
x \boxplus y = xy+1 \\
\text{for all $x, y \in V$. Define an operation of "scalar multiplication" by} \\
a \boxdot x = a^2x \\
\text{for all $a \in R$ and $x \in V$.}
\end{eqnarray}$$
### Expression 10
$$
\left [ \begin{matrix}
0 & -\frac{1}{2} & 0 & \frac{1}{2} & : & 1 & 0 & 0 & 0 \\
0 & 0 & \frac{1}{2} & \frac{1}{2} & : & 0 & 1 & 0 & 0 \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 & : & 0 & 0 & 1 & 0 \\
1 & 0 & \frac{1}{2} & \frac{1}{2} & : & 0 & 0 & 0 & 1 \\
\end{matrix} \right ] \xrightarrow{X_1}
\left [ \begin{matrix}
1 & 0 & \frac{1}{2} & \frac{1}{2} & : & & & & \\
0 & 0 & \frac{1}{2} & \frac{1}{2} & : & & & & \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 & : & & & & \\
0 & -\frac{1}{2} & 0 & \frac{1}{2} & : & & & & \\
\end{matrix} \right ]
$$
$$ \Large
\left [ \begin{matrix}
0 & -\frac{1}{2} & 0 & \frac{1}{2} & : & 1 & 0 & 0 & 0 \\
0 & 0 & \frac{1}{2} & \frac{1}{2} & : & 0 & 1 & 0 & 0 \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 & : & 0 & 0 & 1 & 0 \\
1 & 0 & \frac{1}{2} & \frac{1}{2} & : & 0 & 0 & 0 & 1 \\
\end{matrix} \right ] \xrightarrow{X_1}
\left [ \begin{matrix}
1 & 0 & \frac{1}{2} & \frac{1}{2} & : & & & & \\
0 & 0 & \frac{1}{2} & \frac{1}{2} & : & & & & \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 & : & & & & \\
0 & -\frac{1}{2} & 0 & \frac{1}{2} & : & & & & \\
\end{matrix} \right ]
$$
### Expression 11
$$
a_{jk} = \begin{cases}
0, & \text{for $\vert j-k \vert 1$} \\
1, &\text{for $\vert j-k \vert = 1$} \\
2\cos x, &\text{for $j=k$} \\
\end{cases}
$$
$$ \Huge
a_{jk} = \begin{cases}
0, & \text{for $\vert j-k \vert 1$} \\
1, &\text{for $\vert j-k \vert = 1$} \\
2\cos x, &\text{for $j=k$} \\
\end{cases}
$$
### Expression 12
$$
\bf{4.1.3. Definition.} \text{ Let $\bf{x}$ and $\bf{y}$ be nonzero vectors in $\mathbb{R}^n$. Then $\measuredangle{(\bf{x}, \bf{y})}$, the ANGLE between $\bf{x}$ and $\bf{y}$, is defined by} \\
\measuredangle{(\bf{x}, \bf{y})} = \arccos \frac{\langle \bf{x}, \bf{y}\rangle}{\Vert\bf{x} \Vert \Vert\bf{y} \Vert}
$$
$$\begin{eqnarray}
\bf{4.1.3. Definition.} \text{ Let $\bf{x}$ and $\bf{y}$ be nonzero vectors in $\mathbb{R}^n$. Then $\measuredangle{(\bf{x}, \bf{y})}$, the ANGLE between $\bf{x}$ and $\bf{y}$, is defined by} \\
\measuredangle{(\bf{x}, \bf{y})} = \arccos \frac{\langle \bf{x}, \bf{y}\rangle}{\Vert\bf{x} \Vert \Vert\bf{y} \Vert}
\end{eqnarray}$$
### Expression 13
$$
M\cap N \\
M\cup N
$$
$$ \huge \begin{eqnarray}
M\cap N \\
M\cup N
\end{eqnarray} $$
### Expression 14
$$
\int_{0}^{1}t g(t)\,d t=0 \hspace{50pt} \text{and} \hspace{50pt} \int_{0}^{1}t^{4}g(t)\,d t=0
$$
$$ \huge \begin{eqnarray}
\int_{0}^{1}t g(t)\,d t=0 \hspace{50pt} \text{and} \hspace{50pt} \int_{0}^{1}t^{4}g(t)\,d t=0
\end{eqnarray}$$
### Expression 15
$$
\mathbf{y}=\left(-{\frac{1}{2}},{\frac{\sqrt{3}}{2}}\right)
$$
$$ \huge
\mathbf{y}=\left(-{\frac{1}{2}},{\frac{\sqrt{3}}{2}}\right)
$$
## A LaTeX Meme
以下是实际生成的 LaTeX (.tex) 文件：
% Options for packages loaded elsewhere
\PassOptionsToPackage{unicode}{hyperref}
\PassOptionsToPackage{hyphens}{url}
\PassOptionsToPackage{dvipsnames,svgnames,x11names}{xcolor}
%
\documentclass[
letterpaper,
DIV=11,
numbers=noendperiod]{scrartcl}
\usepackage{amsmath,amssymb}
\usepackage{lmodern}
\usepackage{iftex}
\ifPDFTeX
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{textcomp} % provide euro and other symbols
\else % if luatex or xetex
\usepackage{unicode-math}
\defaultfontfeatures{Scale=MatchLowercase}
\defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
\fi
% Use upquote if available, for straight quotes in verbatim environments
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
\IfFileExists{microtype.sty}{% use microtype if available
\usepackage[]{microtype}
\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
}{}
\makeatletter
\@ifundefined{KOMAClassName}{% if non-KOMA class
\IfFileExists{parskip.sty}{%
\usepackage{parskip}
}{% else
\setlength{\parindent}{0pt}
\setlength{\parskip}{6pt plus 2pt minus 1pt}}
}{% if KOMA class
\KOMAoptions{parskip=half}}
\makeatother
\usepackage{xcolor}
\setlength{\emergencystretch}{3em} % prevent overfull lines
\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
% Make \paragraph and \subparagraph free-standing
\ifx\paragraph\undefined\else
\let\oldparagraph\paragraph
\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}}
\fi
\ifx\subparagraph\undefined\else
\let\oldsubparagraph\subparagraph
\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}}
\fi
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}\usepackage{longtable,booktabs,array}
\usepackage{calc} % for calculating minipage widths
% Correct order of tables after \paragraph or \subparagraph
\usepackage{etoolbox}
\makeatletter
\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{}
\makeatother
% Allow footnotes in longtable head/foot
\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
\makesavenoteenv{longtable}
\usepackage{graphicx}
\makeatletter
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth\else\Gin@nat@width\fi}
\def\maxheight{\ifdim\Gin@nat@height>\textheight\textheight\else\Gin@nat@height\fi}
\makeatother
% Scale images if necessary, so that they will not overflow the page
% margins by default, and it is still possible to overwrite the defaults
% using explicit options in \includegraphics[width, height, ...]{}
\setkeys{Gin}{width=\maxwidth,height=\maxheight,keepaspectratio}
% Set default figure placement to htbp
\makeatletter
\def\fps@figure{htbp}
\makeatother
\KOMAoption{captions}{tableheading}
\makeatletter
\makeatother
\makeatletter
\makeatother
\makeatletter
\@ifpackageloaded{caption}{}{\usepackage{caption}}
\AtBeginDocument{%
\ifdefined\contentsname
\renewcommand*\contentsname{Table of contents}
\else
\newcommand\contentsname{Table of contents}
\fi
\ifdefined\listfigurename
\renewcommand*\listfigurename{List of Figures}
\else
\newcommand\listfigurename{List of Figures}
\fi
\ifdefined\listtablename
\renewcommand*\listtablename{List of Tables}
\else
\newcommand\listtablename{List of Tables}
\fi
\ifdefined\figurename
\renewcommand*\figurename{Figure}
\else
\newcommand\figurename{Figure}
\fi
\ifdefined\tablename
\renewcommand*\tablename{Table}
\else
\newcommand\tablename{Table}
\fi
}
\@ifpackageloaded{float}{}{\usepackage{float}}
\floatstyle{ruled}
\@ifundefined{c@chapter}{\newfloat{codelisting}{h}{lop}}{\newfloat{codelisting}{h}{lop}[chapter]}
\floatname{codelisting}{Listing}
\newcommand*\listoflistings{\listof{codelisting}{List of Listings}}
\makeatother
\makeatletter
\@ifpackageloaded{caption}{}{\usepackage{caption}}
\@ifpackageloaded{subcaption}{}{\usepackage{subcaption}}
\makeatother
\makeatletter
\@ifpackageloaded{tcolorbox}{}{\usepackage[many]{tcolorbox}}
\makeatother
\makeatletter
\@ifundefined{shadecolor}{\definecolor{shadecolor}{rgb}{.97, .97, .97}}
\makeatother
\makeatletter
\makeatother
\ifLuaTeX
\usepackage{selnolig} % disable illegal ligatures
\fi
\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
\urlstyle{same} % disable monospaced font for URLs
\hypersetup{
pdftitle={Learning LaTeX},
pdfauthor={Anish Goyal},
colorlinks=true,
linkcolor={blue},
filecolor={Maroon},
citecolor={Blue},
urlcolor={Blue},
pdfcreator={LaTeX via pandoc}}
\title{Learning LaTeX}
\usepackage{etoolbox}
\makeatletter
\providecommand{\subtitle}[1]{% add subtitle to \maketitle
\apptocmd{\@title}{\par {\large #1 \par}}{}{}
}
\makeatother
\subtitle{A document containing a collection of mathematical expressions
in LaTeX and a `funny' meme about LaTeX for the Applications of Linear
Algebra class at GSMST.}
\author{Anish Goyal}
\date{1/18/23}
\begin{document}
\maketitle
\ifdefined\Shaded\renewenvironment{Shaded}{\begin{tcolorbox}[enhanced, interior hidden, borderline west={3pt}{0pt}{shadecolor}, breakable, frame hidden, boxrule=0pt, sharp corners]}{\end{tcolorbox}}\fi
\renewcommand*\contentsname{Table of contents}
{
\hypersetup{linkcolor=}
\setcounter{tocdepth}{3}
\tableofcontents
}
\hypertarget{mathematical-expressions}{%
\subsection{Mathematical expressions}\label{mathematical-expressions}}
\hypertarget{expression-1}{%
\subsubsection{Expression 1}\label{expression-1}}
\includegraphics{images/image-1069874928.png}
\begin{verbatim}
$$
\begin{cases}
w + x +y +z=6 \\
w \hspace{90px} +y+z =4 \\
w \hspace{90px}+y \hspace{90px} =2 \\
\end{cases}
$$
\end{verbatim}
\[
\Huge
\begin{cases}
w + x +y +z=6 \\
w \hspace{90px} +y+z =4 \\
w \hspace{90px}+y \hspace{90px} =2 \\
\end{cases}
\]
\hypertarget{expression-2}{%
\subsubsection{Expression 2}\label{expression-2}}
\includegraphics{images/image-1212342519.png}
\begin{verbatim}
$$
k\neq-1, 1 \\ {\frac{1}{k+1}},{\frac{1}{k+1}}
$$
\end{verbatim}
\[\Huge
k\neq-1, 1 \\ {\frac{1}{k+1}},{\frac{1}{k+1}}
\]
\hypertarget{expression-3}{%
\subsubsection{Expression 3}\label{expression-3}}
\includegraphics{images/image-1884990960.png}
\begin{verbatim}
$$
\operatorname{tr}A:=\sum_{j=1}^{n}a_{jj}
$$
\end{verbatim}
\[
\Huge
\operatorname{tr}A:=\sum_{j=1}^{n}a_{jj}
\]
\hypertarget{expression-4}{%
\subsubsection{Expression 4}\label{expression-4}}
\includegraphics{images/image-1983034508.png}
\begin{verbatim}
$$
A(\theta)={\left[
\begin{matrix}
\cos{\theta} & -\sin{\theta} \\
\sin{\theta} & \cos{\theta}
\end{matrix}\right]}
$$
\end{verbatim}
\[ \Huge
A(\theta)={\left[
\begin{matrix}
\cos{\theta} & -\sin{\theta} \\
\sin{\theta} & \cos{\theta}
\end{matrix}\right]}
\]
\hypertarget{expression-5}{%
\subsubsection{Expression 5}\label{expression-5}}
\includegraphics{images/image-1969377765.png}
\begin{verbatim}
$$
A = \left [ \begin{matrix}
0 & a & a^2 & a^3 & a^4 \\
0 & 0 & a & a^2 & a^3 \\
0 & 0 & 0 & a & a^2 \\
0 & 0 & 0 & 0 & a \\
0 & 0 & 0 & 0 & 0
\end{matrix} \right ]
$$
\end{verbatim}
\[ \Huge
A = \left [ \begin{matrix}
0 & a & a^2 & a^3 & a^4 \\
0 & 0 & a & a^2 & a^3 \\
0 & 0 & 0 & a & a^2 \\
0 & 0 & 0 & 0 & a \\
0 & 0 & 0 & 0 & 0
\end{matrix} \right ]
\]
\hypertarget{expression-6}{%
\subsubsection{Expression 6}\label{expression-6}}
\includegraphics{images/image-1950951149.png}
\begin{verbatim}
$$
\textbf{3.1.6. Proposition.} \ Let \ n \in \mathbb{N} \ and \ \textbf{M}_{n \times n} \ be \ the \ collection \ of \ all \ n \times n \ matrices. \ There \ is \ exactly \ one \\ function \\
\mathrm{det}: \textbf{M}_{n \times n} \rightarrow \mathbb{R}\colon A\mapsto\operatorname{det}A
$$
\end{verbatim}
\[
\textbf{3.1.6. Proposition.} \ Let \ n \in \mathbb{N} \ and \ \textbf{M}_{n \times n} \ be \ the \ collection \ of \ all \ n \times n \ matrices. \ There \ is \ exactly \ one \ function \\
\mathrm{det}: \textbf{M}_{n \times n} \rightarrow \mathbb{R}\colon A\mapsto\operatorname{det}A
\]
\hypertarget{expression-7}{%
\subsubsection{Expression 7}\label{expression-7}}
\includegraphics{images/image-2143312636.png}
\begin{verbatim}
$$
\textbf{3.1.12. Proposition.} \ If \ A \in \mathbb{N} \ and \ 1 \le j \le n, \ then \\
\mathrm{det}\ A: \sum_{k=1}^{n}a_{jk}C_{jk}
$$
\end{verbatim}
\[\Large \begin{eqnarray}
\textbf{3.1.12. Proposition.} \ If \ A \in \mathbb{N} \ and \ 1 \le j \le n, \ then \\
\mathrm{det}\ A: \sum_{k=1}^{n}a_{jk}C_{jk}
\end{eqnarray}\]
\hypertarget{expression-8}{%
\subsubsection{Expression 8}\label{expression-8}}
\includegraphics{images/image-1940893679.png}
\begin{verbatim}
$$
\vert a d+b e+c f\vert\leq{\sqrt{a^{2}+b^{2}+c^{2}}}{\sqrt{d^{2}+e^{2}+f^{2}}}
$$
\end{verbatim}
\[ \huge
\vert a d+b e+c f\vert\leq{\sqrt{a^{2}+b^{2}+c^{2}}}{\sqrt{d^{2}+e^{2}+f^{2}}}
\]
\hypertarget{expression-9}{%
\subsubsection{Expression 9}\label{expression-9}}
\includegraphics{images/image-331355199.png}
\begin{verbatim}
$$
\text{Let $V$ be the set of real numbers $x$ such that $x >= 0$. Define an operation of "addition" by} \\
x \boxplus y = xy+1 \\
\text{for all $x, y \in V$. Define an operation of "scalar multiplication" by} \\
a \boxdot x = a^2x \\
\text{for all $a \in R$ and $x \in V$.}
$$
\end{verbatim}
\[\begin{eqnarray}
\text{Let $V$ be the set of real numbers $x$ such that $x >= 0$. Define an operation of "addition" by} \\
x \boxplus y = xy+1 \\
\text{for all $x, y \in V$. Define an operation of "scalar multiplication" by} \\
a \boxdot x = a^2x \\
\text{for all $a \in R$ and $x \in V$.}
\end{eqnarray}\]
\hypertarget{expression-10}{%
\subsubsection{Expression 10}\label{expression-10}}
\includegraphics{images/image-1239672658.png}
\begin{verbatim}
$$
\left [ \begin{matrix}
0 & -\frac{1}{2} & 0 & \frac{1}{2} & : & 1 & 0 & 0 & 0 \\
0 & 0 & \frac{1}{2} & \frac{1}{2} & : & 0 & 1 & 0 & 0 \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 & : & 0 & 0 & 1 & 0 \\
1 & 0 & \frac{1}{2} & \frac{1}{2} & : & 0 & 0 & 0 & 1 \\
\end{matrix} \right ] \xrightarrow{X_1}
\left [ \begin{matrix}
1 & 0 & \frac{1}{2} & \frac{1}{2} & : & & & & \\
0 & 0 & \frac{1}{2} & \frac{1}{2} & : & & & & \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 & : & & & & \\
0 & -\frac{1}{2} & 0 & \frac{1}{2} & : & & & & \\
\end{matrix} \right ]
$$
\end{verbatim}
\[ \Large
\left [ \begin{matrix}
0 & -\frac{1}{2} & 0 & \frac{1}{2} & : & 1 & 0 & 0 & 0 \\
0 & 0 & \frac{1}{2} & \frac{1}{2} & : & 0 & 1 & 0 & 0 \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 & : & 0 & 0 & 1 & 0 \\
1 & 0 & \frac{1}{2} & \frac{1}{2} & : & 0 & 0 & 0 & 1 \\
\end{matrix} \right ] \xrightarrow{X_1}
\left [ \begin{matrix}
1 & 0 & \frac{1}{2} & \frac{1}{2} & : & & & & \\
0 & 0 & \frac{1}{2} & \frac{1}{2} & : & & & & \\
\frac{1}{2} & 0 & -\frac{1}{2} & 0 & : & & & & \\
0 & -\frac{1}{2} & 0 & \frac{1}{2} & : & & & & \\
\end{matrix} \right ]
\]
\hypertarget{expression-11}{%
\subsubsection{Expression 11}\label{expression-11}}
\includegraphics{images/image-966103750.png}
\begin{verbatim}
$$
a_{jk} = \begin{cases}
0, & \text{for $\vert j-k \vert 1$} \\
1, &\text{for $\vert j-k \vert = 1$} \\
2\cos x, &\text{for $j=k$} \\
\end{cases}
$$
\end{verbatim}
\[ \Huge
a_{jk} = \begin{cases}
0, & \text{for $\vert j-k \vert 1$} \\
1, &\text{for $\vert j-k \vert = 1$} \\
2\cos x, &\text{for $j=k$} \\
\end{cases}
\]
\hypertarget{expression-12}{%
\subsubsection{Expression 12}\label{expression-12}}
\includegraphics{images/image-698644403.png}
\begin{verbatim}
$$
\textbf{4.1.3. Definition.} \text{ Let $\textbf{x}$ and $\textbf{y}$ be nonzero vectors in $\mathbb{R}^n$. Then $\measuredangle{(\textbf{x}, \textbf{y})}$, the ANGLE between $\textbf{x}$ and $\textbf{y}$, is defined by} \\
\measuredangle{(\textbf{x}, \textbf{y})} = \arccos \frac{\langle \textbf{x}, \textbf{y}\rangle}{\Vert\textbf{x} \Vert \Vert\textbf{y} \Vert}
$$
\end{verbatim}
\[\begin{eqnarray}
\textbf{4.1.3. Definition.} \text{ Let $\textbf{x}$ and $\textbf{y}$ be nonzero vectors in $\mathbb{R}^n$. Then $\measuredangle{(\textbf{x}, \textbf{y})}$, the ANGLE between $\textbf{x}$ and $\textbf{y}$, is defined by} \\
\measuredangle{(\textbf{x}, \textbf{y})} = \arccos \frac{\langle \textbf{x}, \textbf{y}\rangle}{\Vert\textbf{x} \Vert \Vert\textbf{y} \Vert}
\end{eqnarray}\]
\hypertarget{expression-13}{%
\subsubsection{Expression 13}\label{expression-13}}
\includegraphics{images/image-1340050623.png}
\begin{verbatim}
$$
M\cap N \\
M\cup N
$$
\end{verbatim}
\[ \huge \begin{eqnarray}
M\cap N \\
M\cup N
\end{eqnarray} \]
\hypertarget{expression-14}{%
\subsubsection{Expression 14}\label{expression-14}}
\includegraphics{images/image-341248740.png}
\begin{verbatim}
$$
\int_{0}^{1}t g(t)\,d t=0 \hspace{50pt} \text{and} \hspace{50pt} \int_{0}^{1}t^{4}g(t)\,d t=0
$$
\end{verbatim}
\[ \huge \begin{eqnarray}
\int_{0}^{1}t g(t)\,d t=0 \hspace{50pt} \text{and} \hspace{50pt} \int_{0}^{1}t^{4}g(t)\,d t=0
\end{eqnarray}\]
\hypertarget{expression-15}{%
\subsubsection{Expression 15}\label{expression-15}}
\includegraphics{images/image-1876561767.png}
\begin{verbatim}
$$
\mathbf{y}=\left(-{\frac{1}{2}},{\frac{\sqrt{3}}{2}}\right)
$$
\end{verbatim}
\[ \huge
\mathbf{y}=\left(-{\frac{1}{2}},{\frac{\sqrt{3}}{2}}\right)
\]
\hypertarget{a-latex-meme}{%
\subsection{A LaTeX Meme}\label{a-latex-meme}}
\includegraphics{images/image-220842989.png}
\end{document}
我尝试过使用不同的 pdf 引擎（xelatex 或 latexmk），但无济于事！我尝试过删除方程数组，但仍然不起作用。我还尝试过使用 RStudio 而不是 VS Code 来渲染此文件，但没有成功。
答案1
您收到的错误消息来自嵌套数学环境，例如eqnarray嵌套到其他数学环境（如$$...$$和）中\[...\]，使用过时的字体命令（如\bfkoma 类等）。
但是，您不应该滥用数学环境来输入整个句子，而应该先查看各种定理包，以正确设置定理类部分（例如您的命题）。下面是使用该amsthm包的示例：
---
title: "Learning LaTeX"
author: Anish Goyal
date: January 18, 2023
subtitle: "A document containing a collection of mathematical expressions in LaTeX and a 'funny' meme about LaTeX for the Applications of Linear Algebra class at GSMST."
format:
pdf:
pdf-engine: latexmk
latex-auto-install: true
code-overflow: wrap
toc: true
header-includes:
- \usepackage{amsthm}
- \newtheorem{proposition}{Proposition}
---
\begin{proposition}
If $A \in \mathbb{N}$ and $1 \le j \le n$, then
\begin{equation}
\det A: \sum_{k=1}^{n}a_{jk}C_{jk}
\end{equation}
\end{proposition}
