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


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|(}
\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
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
......@@ -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{argument, of a function}\index{function!argument}
$x\in D$
and output
\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, of a relation}\index{relation!reflexive}
any object is related to itself,
(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, 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.
We can restate the formula in
notation}\index{summation notation!for permutation expansion}
notation}\index{summation notation, for permutation expansion}
\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.
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}%
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]
\index{determinant!Laplace expansion}%
\index{Laplace expansion!computes determinant}
% \index{Laplace expansion!computes determinant}
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}
\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}
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, 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
The largest such power is called the
multiplicity\index{multiplicity, of a root}\index{polynomial!multiplicity}
Finding the roots and factors of a high-degree polynomial can be hard.
......@@ -688,7 +688,7 @@ A \definend{nilpotent matrix}\index{matrix!nilpotent}%
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}
......@@ -2,7 +2,7 @@
% 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.
% (
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}.
\index{magic squares|)}
\index{magic square|)}
......@@ -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
......@@ -1970,7 +1970,8 @@ from $V$ to $\Re^n$.
\,\mapsunder{\text{\small Rep}_B}\,\colvec{v_1 \\ \vdots \\ v_n}
It is well-defined\appendrefs{well-defined}\spacefactor1000 %
It is well-defined\appendrefs{well-defined}\spacefactor1000 %
every \( \vec{v} \)
has one and only one such representation (see
......@@ -2020,6 +2021,7 @@ any $n$-dimensional space is isomorphic to $\Re^n$.
\begin{remark} \label{not:WellDefFcns}
The proof has a sentence
about `well-defined.'
Its point is that to be an isomorphism
......@@ -1037,7 +1037,7 @@ orthogonal to $\vec{\kappa}_2$.
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
......@@ -2,7 +2,7 @@
% 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|)}
% Chapter 1, Topic from _Linear Algebra_ Jim Hefferon
% 2001-Jun-09
\topic{Analyzing 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 @@
% 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}
\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.
......@@ -912,5 +913,5 @@ An easy and interesting application is in \cite{Davies}.
% lower-case letters.
\index{Projective Geometry|)}
\index{projective geometry|)}
......@@ -2,7 +2,8 @@
% 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.
......@@ -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.
......@@ -662,6 +663,7 @@ Here is the session at the prompt.
\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|)}
......@@ -40,7 +40,7 @@
\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.
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
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.
\index{voting paradoxes|)}
\index{voting paradox|)}
......@@ -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}
\( \set{\map{f}{\Re}{\Re} \suchthat f(-x)=-f(x) \text{ for all } x} \).
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 \( x \)}}
and the
\definend{odd}\index{odd functions}
\definend{odd}\index{odd function}
polynomials are the members of this set.
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