X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=examples.tex;h=2d05431e2ceb2fcdcdf3314980f3348929a69ea8;hb=07b10e9bcd59b16af1cd6b143f07f0a3cd56d33d;hp=1d79079ee0ba4c1f5fe09422a8e0f363da81e139;hpb=2398b156d5cec30d4ede3e65ae1c89ad08551447;p=mjotex.git

diff --git a/examples.tex b/examples.tex
index 1d79079..2d05431 100644
--- 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
@@ -176,14 +184,16 @@
     matrices $A$ and $B$ is $\kp{A}{B}$. The adjoint of the operator
     $L$ is $\adjoint{L}$, or if it's a matrix, then its transpose is
     $\transpose{L}$. Its trace is $\trace{L}$, and its spectrum---the
-    set of its eigenvalues---is $\spectrum{L}$. Another matrix-specific
-    concept is the Moore-Penrose pseudoinverse of $L$, denoted by
-    $\pseudoinverse{L}$. Finally, the rank of a matrix $L$ is
-    $\rank{L}$. As far as matrix spaces go, we have the $n$-by-$n$
+    set of its eigenvalues---is $\spectrum{L}$. Another
+    matrix-specific concept is the Moore-Penrose pseudoinverse of $L$,
+    denoted by $\pseudoinverse{L}$. Finally, the rank of a matrix $L$
+    is $\rank{L}$. 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.
+    Cartesian product in this case. A handy way to represent the
+    matrix $A \in \Rn[n \times n]$ whose only non-zero entries are on
+    the diagonal is $\diag{\colvec{A_{11},A_{22},\ldots,A_{nn}}}$.
 
     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
@@ -198,7 +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