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 ... ... @@ -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, ... ...
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 ... ... @@ -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*} % 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}{\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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