mjo-hurwitz.tex: new file for Hurwitz algebras (quaternions and octonions).

[mjotex.git] / examples.tex

diff --git a/examples.tex b/examples.tex
index c3e22e7d96e4660356020cb9d66aeac0074304a5..2d05431e2ceb2fcdcdf3314980f3348929a69ea8 100644 (file)
--- a/examples.tex
+++ b/examples.tex
@@ -38,7 +38,8 @@
     containing the set $\set{x,y,z}$.
 
     If $R$ has a multiplicative identity (that is, a unit) element,
-    then that element is denoted by $\unit{R}$.
+    then that element is denoted by $\unit{R}$. Its additive identity
+    element is $\zero{R}$.
   \end{section}
 
   \begin{section}{Algorithm}
@@ -118,8 +119,10 @@
     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: $\Nn[2]$, $\Zn[2]$,
-    $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. Finally, we have the four standard
-    types of intervals in $\Rn[1]$,
+    $\Qn[2]$, $\Rn[2]$, $\Cn[2]$. The symbols $\Fn[1]$, $\Fn[2]$,
+    et cetera, are available for use with a generic field.
+
+    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 },\\
@@ -169,6 +172,11 @@
     \end{itemize}
   \end{section}
 
+  \begin{section}{Hurwitz}
+    Here lies the Hurwitz algebras, like the quaternions
+    $\quaternions$ and octonions $\octonions$.
+  \end{section}
+
   \begin{section}{Linear algebra}
     The absolute value of $x$ is $\abs{x}$, or its norm is
     $\norm{x}$. The inner product of $x$ and $y$ is $\ip{x}{y}$ and
@@ -200,7 +208,8 @@
     instead.
 
     If you want to solve a system of equations, try Cramer's
-    rule~\cite{ehrenborg}.
+    rule~\cite{ehrenborg}. Or at least the reduced row-echelon form of
+    the matrix, $\rref{A}$.
 
     The direct sum of $V$ and $W$ is $\directsum{V}{W}$, of course,
     but what if $W = V^{\perp}$? Then we wish to indicate that fact by