Commit 2eeba72d authored by Jim Hefferon's avatar Jim Hefferon

index

parent 195c16bd
......@@ -263,7 +263,7 @@ is unique, even thouge no such number exists.)
\appendsection{Techniques of Proof}
\appendsection{Techniques of Proof}\index{proof techniques|(}
\startword{Induction}
\index{induction, mathematical}
......@@ -413,7 +413,7 @@ That's a contradiction, so a rational number with a square of
% Both of these examples aimed to prove something doesn't exist.
% A negative proposition often suggests a proof by contradiction.
\index{proof techniques|)}
......@@ -451,7 +451,7 @@ are so important that their names are reserved, such as the
real numbers \( \Re \)
and the complex numbers \( \C \)).
To denote that something is an
\definend{element\/},\index{element}\index{set!element}
\definend{element},\index{element}\index{set!element}
or \definend{member},\index{member}\index{set!member}) of a set we
use `\(\in \)',
so that \( 7\in\set{3,5,7} \) while \( 8\not\in\set{3,5,7} \).
......@@ -459,7 +459,7 @@ so that \( 7\in\set{3,5,7} \) while \( 8\not\in\set{3,5,7} \).
We say that \( A \) is a \definend{subset} of \( B \), written
$A\subseteq B$, when $x\in A$ implies that $x\in B$.
In this book we use
`\( \subset \)' for the \definend{proper subset}\index{sets!proper subset}\index{proper!subset}\index{sets!subset} %
`\( \subset \)' for the \definend{proper subset}\index{sets!proper subset}\index{proper subset}\index{sets!subset}\index{subset, proper} %
relationship that \( A \) is a subset of \( B \) but \( A\neq B \)
(some authors use this symbol for any kind of subset, proper or not).
An example is
......@@ -507,7 +507,7 @@ It shows that if \( x\in P \) then \( x\in Q \).
\startword{Set Operations}
The \definend{union}\index{union}\index{set!union} of two sets is
The \definend{union}\index{union of sets}\index{set!union} of two sets is
\( P\union Q=\set{x\suchthat \text{$(x\in P)$ or $(x\in Q)$}} \).
The diagram shows that an element is in the union if it is in either of the
sets.
......@@ -552,7 +552,7 @@ not collapse.
These are \definend{sequences},\index{sequence} denoted with angle brackets:
\( \sequence{ 2,3,7}\neq\sequence{2,7,3} \).
A sequence of length \( 2 \) is an
\definend{ordered pair},\index{ordered pair}\index{pair!ordered}
\definend{ordered pair},\index{ordered pair}\index{pair, ordered}
and is often written with parentheses: \( (\pi,3) \).
We also sometimes say `ordered triple', `ordered \( 4 \)-tuple', etc.
The set of ordered \( n \)-tuples of elements of a set \( A \) is denoted
......@@ -567,10 +567,10 @@ Thus \( \Re^2 \) is the set of pairs of reals.
A \definend{function}\index{function}
or \definend{map}\index{map} $\map{f}{D}{C}$ is
is an association between input
\definend{arguments}\index{function!argument}
\definend{arguments}\index{argument, of a function}\index{function!argument}
$x\in D$
and output
\definend{values}\index{value}\index{function!value}
\definend{values}\index{value, of a function}\index{function!value}
$f(x)\in C$ subject to the the requirement
that the
function must be \definend{well-defined},\index{function!well-defined}%
......@@ -769,12 +769,12 @@ A binary relation \( \set{(a,b),\ldots } \)
is an
\definend{equivalence relation}\index{equivalence!relation}\index{relation!equivalence}
when it satisfies
(1)~\definend{reflexivity}:\index{reflexivity}\index{relation!reflexive}
(1)~\definend{reflexivity}:\index{reflexivity, of a relation}\index{relation!reflexive}
any object is related to itself,
(2)~\definend{symmetry}:\index{symmetry}\index{relation!symmetric}
(2)~\definend{symmetry}:\index{symmetry, of a relation}\index{relation!symmetric}
if \( a \) is related to \( b \) then
\( b \) is related to \( a \), and
(3)~\definend{transitivity}:\index{transitivity}\index{relation!transitive}
(3)~\definend{transitivity}:\index{transitivity, of a relation}\index{relation!transitive}
if \( a \) is related to \( b \) and \( b \) is
related to \( c \) then \( a \) is related to \( c \).
Some examples (on the integers): `\( = \)' is an equivalence relation,
......
No preview for this file type
......@@ -2545,7 +2545,7 @@ where \( \phi_1,\ldots,\phi_k \) are all of the \( n \)-permutations.
%<*SummationForPermutationExpansion>
We can restate the formula in
\definend{summation
notation}\index{summation notation!for permutation expansion}
notation}\index{summation notation, for permutation expansion}
\begin{equation*}
\deter{T}=\!\!
\sum_{\text{permutations\ }\phi}\!\!\!\!
......
......@@ -11,7 +11,7 @@ to compute determinants by hand.
\subsectionoptional{Laplace's Expansion}
\index{Laplace expansion|(}
\index{Laplace determinant expansion|(}
% We can compute
% the determinant of an~$\nbyn{n}$ matrix as a combination
......@@ -175,7 +175,7 @@ case in terms of smaller ones.
%<*df:Minor>
For any \( \nbyn{n} \) matrix \( T \), the \( \nbyn{(n-1)} \) matrix formed by
deleting row~\( i \) and column~\( j \) of \( T \) is the
\( i,j \) \definend{minor}\index{minor}\index{determinant!minor}%
\( i,j \) \definend{minor}\index{minor, of a matrix}\index{determinant!minor}%
\index{matrix!minor}
of \( T \).
The \( i,j \) \definend{cofactor}\index{cofactor}\index{determinant!using cofactors}%
......@@ -226,7 +226,7 @@ these are the \( 1,2 \) and \( 2,2 \) cofactors.
\begin{theorem}[Laplace Expansion of Determinants]
\label{th:LaPlaceExp}
\index{determinant!Laplace expansion}%
\index{Laplace expansion!computes determinant}
% \index{Laplace expansion!computes determinant}
%<*th:LaPlaceExp>
Where \( T \) is an \( \nbyn{n} \) matrix, we can find
the determinant by expanding by cofactors on any
......@@ -1269,4 +1269,4 @@ Gauss-Jordan method.
% =\deter{\adj(T)}\cdot \deter{T}$.
% \end{answer}
\end{exercises}
\index{Laplace expansion|)}
\index{Laplace determinant expansion|)}
......@@ -279,7 +279,7 @@ translation is a \definend{glide reflection}\index{reflection!glide}).
Another idea encountered in
elementary geometry, besides congruence of figures, is
that figures are
\definend{similar}\index{similar triangles}\index{triangles!similar}
\definend{similar}\index{similar triangles}\index{triangles, similar}
if they are congruent after a change of scale.
The two triangles below are similar since the second is
the same shape as the first but $3/2$-ths the size.
......
......@@ -844,7 +844,7 @@ generate.
\begin{definition} \label{df:Length}
%<*df:Length>
The \definend{length}\index{vector!length}\index{length}
The \definend{length}\index{vector!length}\index{length of a vector}
(or \definend{norm}\index{vector!norm}\index{norm})
of a vector
\( \vec{v}\in\Re^n \) is the square root of the sum of the squares of its
......@@ -863,7 +863,7 @@ A classic motivating discussion is in \cite{MathematicsPlausReason}.
Note that for any nonzero $\vec{v}$, the vector $\vec{v}/\norm{\vec{v}}$
has length one.
We say that the second
\definend{normalizes}\index{normalize}\index{vector!normalize}
\definend{normalizes}\index{normalize, vector}\index{vector!normalize}
$\vec{v}$ to length one.
We can use that to get a formula
......
......@@ -190,7 +190,7 @@ A repeated root of a polynomial is a number $\lambda$ such that
the polynomial is evenly divisible by $(x-\lambda)^n$ for some power larger than
one.
The largest such power is called the
multiplicity\index{multiplicity}\index{polynomial!multiplicity}
multiplicity\index{multiplicity, of a root}\index{polynomial!multiplicity}
of~$\lambda$.
Finding the roots and factors of a high-degree polynomial can be hard.
......
......@@ -688,7 +688,7 @@ A \definend{nilpotent matrix}\index{matrix!nilpotent}%
\index{nilpotent!matrix}
is one with a power that is the zero matrix.
In either case, the least such power is the \definend{index of nilpotency}.%
\index{nilpotency!index}\index{index, of nilpotency}
\index{nilpotency, index of}\index{index of nilpotency}
\end{definition}
\begin{example}
......
......@@ -2,7 +2,7 @@
% http://joshua.smcvt.edu/linearalgebra
% 2012-Feb-12
\topic{Magic Squares}
\index{magic squares|(}
\index{magic square|(}
A Chinese legend tells the story of a
flood by the Lo river.
People offered sacrifices to appease the river.
......@@ -33,7 +33,7 @@ the river's anger cooled.
% (http://en.wikipedia.org/wiki/Lo_Shu_Square)
A square matrix is
\definend{magic}\index{magic square}\index{matrix!magic square}
\definend{magic}\index{matrix!magic square}\index{magic square!definition}
if each row, column, and diagonal add to the same
number, the matrix's \definend{magic number}.
......@@ -574,7 +574,7 @@ The proof given here began with \cite{Ward}.
\end{exparts}
\end{answer}
\end{exercises}
\index{magic squares|)}
\index{magic square|)}
\endinput
......@@ -1867,7 +1867,7 @@ Thus the composition is an isomorphism.
Since it is an equivalence,
isomorphism partitions the universe of vector spaces
into classes:\index{partitions!into isomorphism classes}
into classes:\index{partition!into isomorphism classes}
each space is in one and only one isomorphism class.
\begin{center} \small
\raisebox{.5in}{\begin{tabular}{l}
......@@ -1970,7 +1970,8 @@ from $V$ to $\Re^n$.
\,\mapsunder{\text{\small Rep}_B}\,\colvec{v_1 \\ \vdots \\ v_n}
\end{equation*}
%</pf:EqDimImpIso1>
It is well-defined\appendrefs{well-defined}\spacefactor1000 %
It is well-defined\appendrefs{well-defined}\spacefactor1000 %
\index{well-defined}
since
every \( \vec{v} \)
has one and only one such representation (see
......@@ -2020,6 +2021,7 @@ any $n$-dimensional space is isomorphic to $\Re^n$.
\end{proof}
\begin{remark} \label{not:WellDefFcns}
\index{well-defined}
The proof has a sentence
about `well-defined.'
Its point is that to be an isomorphism
......
......@@ -1037,7 +1037,7 @@ orthogonal to $\vec{\kappa}_2$.
\end{proof}
In addition to having the vectors in the basis be orthogonal, we can also
\definend{normalize}\index{normalize} each vector by dividing
\definend{normalize}\index{normalize, vector} each vector by dividing
by its length, to end with an
\definend{orthonormal basis}.\index{basis!orthonormal}%
\index{orthonormal basis}.
......@@ -2255,7 +2255,7 @@ it is projection along a subspace perpendicular to the line.
\begin{definition} \label{def:OrthComp}
The \definend{orthogonal complement\/}\index{orthogonal!complement}%
\index{complementary subspaces!orthogonal}\index{perp}
\index{complementary subspaces!orthogonal}\index{perp, of a subspace}
of a subspace \( M \) of $\Re^n$ is
\begin{equation*}
M^\perp=\set{\vec{v}\in\Re^n\suchthat
......
......@@ -2,7 +2,7 @@
% http://joshua.smcvt.edu/linearalgebra
% 2001-Jun-12
\topic{Markov Chains}
\index{Markov chains|(}
\index{Markov chain|(}
Here is a simple game:
a player bets on coin tosses, a dollar each time,
......@@ -134,7 +134,7 @@ the game is over.
% (Because a player who enters either of the boundary states never leaves, they
% are said to be \definend{absorbtive}.)\index{state!absorbtive}
This is a \definend{Markov chain}.\index{Markov chain}
This is a \definend{Markov chain}.\index{Markov chain!definition}
Each vector is a
\definend{probability vector},\index{vector!probability}%
\index{probability vector} whose entries are nonnegative real numbers that
......@@ -2229,6 +2229,6 @@ octave:6> gplot z
% \noindent Translating to another computer algebra system should be
% easy\Dash all
% have commands similar to these.
\index{Markov chains|)}
\index{Markov chain|)}
\endinput
% Chapter 1, Topic from _Linear Algebra_ Jim Hefferon
% http://joshua.smcvt.edu/linalg.html
% http://joshua.smcvt.edu/linearalgebra
% 2001-Jun-09
\topic{Analyzing Networks}
\index{networks|(}
......@@ -22,8 +22,8 @@ and two facts about electrical networks.
The first fact is that a battery is like a pump,
providing a force impelling the electricity to flow, if there is a path.
We say that the battery provides a \definend{potential}\index{potential}
to flow.
We say that the battery provides a
\definend{potential}.\index{potential, electric}
% Of course, this network accomplishes its function when, as the electricity
% flows, it goes through a light.
For instance, when the driver steps on the brake then the switch makes contact
......
......@@ -2,7 +2,7 @@
% http://joshua.smcvt.edu/linearalgebra
% 2001-Jun-12
\topic{Projective Geometry}
\index{Projective Geometry|(}
\index{projective geometry|(}
There are geometries other than the familiar Euclidean one.
One such geometry arose when artists observed
that what a viewer sees is not necessarily what is there.
......@@ -289,7 +289,8 @@ $1L_1+2L_2+3L_3=0$
and so this is the equation of $v$.
This symmetry of the statements about lines and points is
the \definend{Duality Principle}\index{projective geometry!Duality Principle}
the
\definend{Duality Principle}\index{Duality Principle, of projective geometry}
of projective geometry:~in any true statement,
interchanging `point' with `line' results in another true statement.
For example, just as two distinct points determine one and only one line,
......@@ -378,7 +379,7 @@ dependent.
The following result is more evidence of the niceness
of the geometry of the projective plane.
These two triangles are
\definend{in perspective}\index{perspective!triangles} from the point $O$
\definend{in perspective}\index{perspective, triangles} from the point $O$
because their corresponding vertices are collinear.
\begin{center}
\includegraphics{ch4.19}
......@@ -912,5 +913,5 @@ An easy and interesting application is in \cite{Davies}.
% lower-case letters.
\end{answer}
\end{exercises}
\index{Projective Geometry|)}
\index{projective geometry|)}
\endinput
......@@ -2,7 +2,8 @@
% http://joshua.smcvt.edu/linearalgebra
% 2001-Jun-12
\topic{Linear Recurrences}
\index{linear recurrences|(}
\index{linear recurrence|(}
\index{recurrence relation|(}
In 1202 Leonardo of Pisa, known as Fibonacci, posed this problem.
\begin{quotation}
......@@ -156,7 +157,7 @@ before the onset of fertility, we nonetheless
still get a function that is asymptotically exponential.
In general, a
\definend{homogeneous linear recurrence relation of order $k$}\index{recurrence}\index{linear recurrence}
\definend{homogeneous linear recurrence relation of order $k$}\index{recurrence}\index{linear recurrence!definition}
has this form.
\begin{equation*}
f(n)=a_{n-1}f(n-1)+a_{n-2}f(n-2)+\dots+a_{n-k}f(n-k)
......@@ -662,6 +663,7 @@ Here is the session at the prompt.
\end{lstlisting}
\noindent This is a list of $T(1)$ through $T(64)$ (the session was
edited to put in line breaks for readability).
\index{linear recurrences|)}
\index{recurrence relation|)}
\index{linear recurrence|)}
\endinput
......@@ -40,7 +40,7 @@
%\newcommand{\votinggraphic}[1]{\hspace{.8em}\mathord{\raisebox{-.2in}[.3in][.2in]{\includegraphics{voting.#1}}}\hspace{.8em}}
\topic{Voting Paradoxes}
\index{voting paradoxes|(}
\index{voting paradox|(}
%\emph{(Optional material from this chapter is discussed here,
%but is not required to complete the exercises.)}
......@@ -96,7 +96,7 @@ prefers the Third to the Democrat, eighteen to eleven.
\begin{center}
\includegraphics{voting.1}
\end{center}
This is a \definend{voting paradox}\index{voting paradox},
This is a \definend{voting paradox}\index{voting paradox!definition},
specifically, a
\definend{majority cycle}.\index{voting paradox!majority cycle}
......@@ -246,7 +246,7 @@ This voter's decomposition
\end{equation*}
shows that these opposite preferences have decompositions that are opposite.
We say that the first voter has positive
\definend{spin}\index{spin}\index{voting paradoxes!spin}
\definend{spin}\index{spin}\index{voting paradox!spin}
since the cycle part is with the direction
that we have chosen for the arrows,
while the second voter's spin is negative.
......@@ -921,7 +921,7 @@ for his kind and illuminating discussions.)}
This holds if $U$ is any subset, subspace or not.
\end{answer}
\end{exercises}
\index{voting paradoxes|)}
\index{voting paradox|)}
\endinput
......
......@@ -3114,7 +3114,7 @@ The next section studies spanning sets that are minimal.
\( \set{\map{f}{\Re}{\Re} \suchthat f(-x)=f(x) \text{ for all } x} \).
For example, two members of this set are $f_1(x)=x^2$
and $f_2(x)=\cos (x)$.
\partsitem The \definend{odd}\index{function!odd}\index{odd functions}
\partsitem The \definend{odd}\index{function!odd}\index{odd function}
functions
\( \set{\map{f}{\Re}{\Re} \suchthat f(-x)=-f(x) \text{ for all } x} \).
Two members are $f_3(x)=x^3$ and $f_4(x)=\sin(x)$.
......
......@@ -4915,7 +4915,7 @@ at the end of the Chapter Five.
\set{p\in\polyspace_n \suchthat \text{\( p(-x)=p(x) \) for all \( x \)}}
\end{equation*}
and the
\definend{odd}\index{odd functions}
\definend{odd}\index{odd function}
polynomials are the members of this set.
\begin{equation*}
\mathcal{O}=
......
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