superscript when that superscript would be one: $\Nn[1]$,
$\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
superscript is (say) two, then it appears: $\Nn[2]$, $\Zn[2]$,
- $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. Likewise we have the $n$-by-$n$
- real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
- respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
- simplify because the ``$n$'' does not indicate the arity of a
- Cartesian product in this case.
+ $\Qn[2]$, $\Rn[2]$, $\Cn[2]$.
We also have a few basic set operations, for example the union of
two or three sets: $\union{A}{B}$, $\unionthree{A}{B}{C}$. And of
\unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
\end{equation*}
%
- The powerset of $X$ displays nicely, as $\powerset{X}$. Finally,
- we have the four standard types of intervals in $\Rn[1]$,
+ Finally, we have the four standard types of intervals in $\Rn[1]$,
%
\begin{align*}
\intervaloo{a}{b} &= \setc{ x \in \Rn[1]}{ a < x < b },\\
$\transpose{L}$. Its trace is $\trace{L}$. Another matrix-specific
concept is the Moore-Penrose pseudoinverse of $L$, denoted by
$\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
- $\rank{L}$.
+ $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
+ real-symmetric and complex-Hermitian matrices $\Sn$ and $\Hn$
+ respectively; however $\Sn[1]$ and $\Hn[1]$ do not automatically
+ simplify because the ``$n$'' does not indicate the arity of a
+ Cartesian product in this case.
The span of a set $X$ is $\spanof{X}$, and its codimension is
$\codim{X}$. The projection of $X$ onto $V$ is $\proj{V}{X}$. The
system to test them.
\end{section}
- \begin{section}{Miscellaneous}
- The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
- = 3$.
- \end{section}
-
\begin{section}{Proof by cases}
\begin{proposition}
\renewcommand{\baselinestretch}{1}
\end{section}
+ \begin{section}{Set theory}
+ The cardinality of the set $X \coloneqq \set{1,2,3}$ is $\card{X}
+ = 3$, and its powerset is $\powerset{X}$.
+ \end{section}
+
\begin{section}{Theorems}
\begin{corollary}
The