mjo-linear_algebra.tex: add the Moore-Penrose \pseudoinverse.

[mjotex.git] / examples.tex

diff --git a/examples.tex b/examples.tex
index 98ddfd550b08ea4fd78301eb47c0aea19cb8de7a..f33ed747d4f5e03dd3cb33f7106e9a417d8f2748 100644 (file)
--- a/examples.tex
+++ b/examples.tex
@@ -28,20 +28,29 @@
   \begin{section}{Arrow}
     The identity operator on $V$ is $\identity{V}$. The composition of
     $f$ and $g$ is $\compose{f}{g}$. The inverse of $f$ is
-    $\inverse{f}$.
+    $\inverse{f}$. If $f$ is a function and $A$ is a subset of its
+    domain, then the preimage under $f$ of $A$ is $\preimage{f}{A}$.
   \end{section}
 
   \begin{section}{Common}
     The function $f$ applied to $x$ is $f\of{x}$. We can group terms
     like $a + \qty{b - c}$ or $a + \qty{b - \sqty{c - d}}$. Here's a
     set $\set{1,2,3} = \setc{n \in \Nn[1]}{ n \le 3 }$. Here's a pair
-    of things $\pair{1}{2}$ or a triple of them
-    $\triple{1}{2}{3}$. The Cartesian product of two sets $A$ and $B$
-    is $\cartprod{A}{B}$; if we take the product with $C$ as well,
-    then we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$
-    and $W$ is $\directsum{V}{W}$ and the factorial of the number $10$
-    is $\factorial{10}$.
-
+    of things $\pair{1}{2}$ or a triple of them $\triple{1}{2}{3}$,
+    and the factorial of the number $10$ is $\factorial{10}$.
+
+    The Cartesian product of two sets $A$ and $B$ is
+    $\cartprod{A}{B}$; if we take the product with $C$ as well, then
+    we obtain $\cartprodthree{A}{B}{C}$. The direct sum of $V$ and $W$
+    is $\directsum{V}{W}$. Or three things,
+    $\directsumthree{U}{V}{W}$. How about more things? Like
+    $\directsummany{k=1}{\infty}{V_{k}} \ne
+    \cartprodmany{k=1}{\infty}{V_{k}}$. Those direct sums and
+    cartesian products adapt nicely to display equations:
+    %
+    \begin{equation*}
+      \directsummany{k=1}{\infty}{V_{k}} \ne \cartprodmany{k=1}{\infty}{V_{k}}.
+    \end{equation*}
     Here are a few common tuple spaces that should not have a
     superscript when that superscript would be one: $\Nn[1]$,
     $\Zn[1]$, $\Qn[1]$, $\Rn[1]$, $\Cn[1]$. However, if the
@@ -60,7 +69,6 @@
     \begin{equation*}
       \unionmany{k=1}{\infty}{A_{k}} = \intersectmany{k=1}{\infty}{B_{k}}
     \end{equation*}
-    %
   \end{section}
 
   \begin{section}{Cone}
@@ -260,5 +268,5 @@
     The interior of a set $X$ is $\interior{X}$. Its closure is
     $\closure{X}$ and its boundary is $\boundary{X}$.
   \end{section}
-  
+
 \end{document}