From 195c16bd7aa655cf728092614ee9e34d3eea0aba Mon Sep 17 00:00:00 2001 From: Jim Hefferon Date: Wed, 27 Nov 2013 07:55:39 -0500 Subject: [PATCH] index through l --- appen.tex | 29 +++++++++++++++-------------- book.pdf | Bin 7428897 -> 7428922 bytes cramer.tex | 2 +- crystal.tex | 6 +++--- det3.tex | 4 ++-- dimen.tex | 6 +++--- erlang.tex | 8 ++++---- gr1.tex | 10 ++++++---- gr2.tex | 7 ++++--- gr3.tex | 2 +- homogeom.tex | 3 ++- jc1.tex | 13 +++++++------ jc2.tex | 6 +++--- jc3.tex | 4 ++-- jc4.tex | 4 ++-- map2.tex | 12 ++++++------ map4.tex | 4 ++-- markov.tex | 2 +- ppivot.tex | 2 +- projplane.tex | 4 ++-- vs1.tex | 3 +-- vs3.tex | 10 +++++----- 22 files changed, 73 insertions(+), 68 deletions(-) diff --git a/appen.tex b/appen.tex index 34d8665..52a57b3 100644 --- a/appen.tex +++ b/appen.tex @@ -28,12 +28,12 @@ Two other sources, available online, are %\bigskip %\par\noindent Formal mathematical statements come labelled as a -\definend{Theorem}\index{theorem} +\definend{Theorem} % \index{theorem} for major points, -a \definend{Corollary}\index{corollary} +a \definend{Corollary} % \index{corollary} for results that follow immediately from a prior one, or a -\definend{Lemma}\index{lemma} +\definend{Lemma} %\index{lemma} for results chiefly used to prove others. Statements can be complex and have many parts. @@ -266,7 +266,7 @@ is unique, even thouge no such number exists.) \appendsection{Techniques of Proof} \startword{Induction} -\index{induction} +\index{induction, mathematical} Many proofs are iterative, Here's why the statement is true for the number $$0$$, it then follows for $$1$$ and from there to $$2$$ \ldots''. @@ -285,12 +285,12 @@ Our induction proofs involve statements with one free natural number variable. Each proof has two steps. -In the \definend{base step}\index{base step!of induction} +In the \definend{base step}\index{base step, of induction proof} we show that the statement holds for some intial number $i\in \N$. Often this step is a routine, and short, verification. The second step, -the \definend{inductive step},\index{inductive step!of induction} +the \definend{inductive step},\index{inductive step, of induction proof} is more subtle; we will show that this implication holds: \begin{equation*} \begin{tabular}{l} @@ -514,7 +514,7 @@ sets. \begin{center} \includegraphics{appen.3} \end{center} -The \definend{intersection}\index{intersection}\index{set!intersection} is +The \definend{intersection}\index{intersection, of sets}\index{set!intersection} is $$P\intersection Q=\set{x\suchthat \text{(x\in P) and (x\in Q)}}$$. \begin{center} \includegraphics{appen.2} @@ -567,7 +567,7 @@ 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}\index{function!argument} +\definend{arguments}\index{function!argument} $x\in D$ and output \definend{values}\index{value}\index{function!value} @@ -608,7 +608,7 @@ We often use $y$ to denote $f(x)$. We also use the notation $$x\mapsunder{f} 16x^2-100$$, read $$x$$ maps under $$f$$ to $$16x^2-100$$' or $$16x^2-100$$ is the -\definend{image}\index{image!under a function}\index{function!image} +\definend{image}\index{image, under a function}\index{function!image} of $$x$$'. A map such as $$x\mapsto \sin(1/x)$$ is a @@ -632,11 +632,12 @@ that the number $$0$$ plays in real number addition or that $$1$$ plays in multiplication. In line with that analogy, we define a -\definend{left inverse}\index{inverse!left} of a map +\definend{left inverse}\index{inverse!function!left}\index{inverse!left}\index{left inverse} of a map $$\map{f}{X}{Y}$$ to be a function $$\map{g}{\text{range}(f)}{X}$$ such that $$\composed{g}{f}$$ is the identity map on $$X$$. -A \definend{right inverse}\index{inverse!right} of $$f$$ is a +A \definend{right inverse}\index{inverse!function!right}\index{inverse!right}\index{right inverse} +of $$f$$ is a $$\map{h}{Y}{X}$$ such that $$\composed{f}{h}$$ is the identity. For some $f$'s there is a map that is @@ -648,7 +649,7 @@ If such a map exists then it is unique because if both $$g_1$$ and =g_2(x) \) (the middle equality comes from the associativity of function composition) so we call it a \definend{two-sided inverse} or just -\definend{the'' inverse},\index{inverse}\index{inverse!two-sided}\index{function!inverse} +\definend{the'' inverse},\index{inverse}\index{inverse!two-sided}\index{function!inverse}\index{inverse!function}\index{inverse function}\index{inversion} and denote it $$f^{-1}$$. For instance, the inverse of the function $$\map{f}{\Re}{\Re}$$ given by $$f(x)=2x-3$$ is the function $$\map{f^{-1}}{\Re}{\Re}$$ @@ -759,7 +760,7 @@ are covered. \startword{Equivalence Relations} -\index{relation!equivalence}\index{equivalence relation} +\index{relation!equivalence}\index{equivalence relation}\index{equivalence} We shall need to express that two objects are alike in some way. 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The generalization of this example is \definend{Cramer's Rule}:% \index{determinant!Cramer's rule}% -\index{linear equation!solutions of!Cramer's rule} +\index{linear equation!solution of!Cramer's rule} if $$\deter{A}\neq 0$$ then the system $$A\vec{x}=\vec{b}$$ has the unique solution $diff --git a/crystal.tex b/crystal.tex index cce0de4..dddc321 100644 --- a/crystal.tex +++ b/crystal.tex @@ -3,7 +3,7 @@ % 2001-Jun-11 \topic{Crystals} \index{crystals|(} -Everyone has noticed that table salt\index{crystals!salt}\index{salt} +Everyone has noticed that table salt\index{salt} comes in little cubes. \begin{center} \includegraphics[height=1.25in]{salt.jpg} %1.25in tall @@ -43,7 +43,7 @@ Then we can describe, say, the corner in the upper right of the picture above as$3\vec{\beta}_1+2\vec{\beta}_2$. Another crystal from everyday experience is pencil lead. -It is \definend{graphite},\index{crystals!graphite} +It is \definend{graphite},\index{graphite} formed from carbon atoms arranged in this shape. \begin{center} %graphite \includegraphics{ch2.10} @@ -72,7 +72,7 @@ so this \tag*{}\end{equation*} is a good basis. -Another familiar crystal formed from carbon is diamond.\index{crystals!diamond} +Another familiar crystal formed from carbon is diamond.\index{diamond} Like table salt it is built from cubes but the structure inside each cube is more complicated. In addition to carbons at each corner, diff --git a/det3.tex b/det3.tex index 058bc54..f3aa6ae 100644 --- a/det3.tex +++ b/det3.tex @@ -178,8 +178,8 @@ deleting row~$$i$$ and column~$$j$$ of $$T$$ is the $$i,j$$ \definend{minor}\index{minor}\index{determinant!minor}% \index{matrix!minor} of $$T$$. -The $$i,j$$ \definend{cofactor}\index{cofactor}\index{determinant!cofactor}% -\index{matrix!cofactor} +The $$i,j$$ \definend{cofactor}\index{cofactor}\index{determinant!using cofactors}% +% \index{matrix!cofactor} $$T_{i,j}$$ of $$T$$ is $$(-1)^{i+j}$$ times the determinant of the $$i,j$$ minor of $$T$$. % diff --git a/dimen.tex b/dimen.tex index 440b947..ab1920a 100644 --- a/dimen.tex +++ b/dimen.tex @@ -26,7 +26,7 @@ However it is not correct in other unit systems, because$16$isn't the right constant in those systems. We can fix that by attaching units to the$16$, making it a -\definend{dimensional constant}\index{dimensional constant}. +\definend{dimensional constant}\index{dimensional!constant}. \begin{equation*} \text{dist}=16\,\frac{\text{ft}}{\text{sec}^2}\cdot (\text{time})^2 \end{equation*} @@ -48,12 +48,12 @@ Moving away from a specific unit system allows us to just say that we measure all quantities here in combinations of some units of length~$L$, mass~$M$, and time~$T$. These three are our -\definend{dimensions}\index{dimension!physical}. +\definend{physical dimensions}\index{physical dimension}. For instance, we could measure velocity in$\text{feet}/\text{second}$or$\text{fathoms}/\text{hour}$but at all events it involves a unit of length divided by a unit of time -so the \definend{dimensional formula}\index{dimensional formula} +so the \definend{dimensional formula}\index{dimensional!formula} of velocity is$L/T$. Similarly, we could state density's dimensional formula as$M/L^3$. diff --git a/erlang.tex b/erlang.tex index bad1500..e7657ba 100644 --- a/erlang.tex +++ b/erlang.tex @@ -8,9 +8,9 @@ In \emph{The Elements},\index{Euclid} Euclid considers two figures to be the same if they have the same size and shape. That is, while the triangles below are not equal because they are not the same set of points, -they are \definend{congruent}\index{congruent figures}\Dash essentially +they are, for Euclid's purposes, essentially indistinguishable -for Euclid's purposes\Dash because we can imagine +because we can imagine picking the plane up, sliding it over and rotating it a bit, although not warping or stretching it, @@ -27,8 +27,8 @@ map from the plane to itself. Euclid considers only transformations that may slide or turn the plane but not bend or stretch it. Accordingly, define a map$\map{f}{\Re^2}{\Re^2}$to be -\definend{distance-preserving}\index{distance-preserving}% -\index{map!distance-preserving} +\definend{distance-preserving}\index{distance-preserving map}% +\index{map!distance-preserving}\index{function!distance-preserving} or a \definend{rigid motion}\index{rigid motion} or an \definend{isometry}\index{isometry} if for all points$P_1,P_2\in\Re^2$, diff --git a/gr1.tex b/gr1.tex index 9391edd..835c3a7 100644 --- a/gr1.tex +++ b/gr1.tex @@ -145,7 +145,7 @@ This algorithm is or \definend{linear elimination}\index{linear elimination}% \index{system of linear equations!linear elimination}% \index{system of linear equations!elimination}% -\index{elimination}). +\index{elimination, Gaussian}). % It transforms the system, step by step, into one % with a form that we can easily solve. % We will first illustrate how it goes and then we will see the @@ -1912,7 +1912,7 @@ is a rectangular array of numbers with $$m$$~\definend{rows}\index{matrix!row}\index{row} and $$n$$~\definend{columns}\index{matrix!column}\index{column}. Each number in the matrix is an -\definend{entry}\index{matrix!entry}\index{entry}. +\definend{entry}\index{matrix!entry}\index{entry, matrix}. % \end{definition} @@ -2021,7 +2021,7 @@ is a matrix with a single column. A matrix with a single row is a \definend{row vector}\index{row!vector}\index{vector!row}. The entries of a vector are its -\definend{components}\index{component}\index{vector!component}. +\definend{components}\index{component of a vector}\index{vector!component}. A column or row vector whose components are all zeros is a \definend{zero vector}.\index{zero vector}\index{vector!zero} % @@ -2069,7 +2069,9 @@ we first need to define these operations. \begin{definition} \label{df:VectorSum} %<*df:VectorSum> -The \definend{vector sum}\index{vector!sum}\index{sum!vector} of +The +\definend{vector sum}\index{vector!sum}\index{sum!vector}\index{addition of vectors} +of $$\vec{u}$$ and $$\vec{v}$$ is the vector of the sums. \begin{equation*} \vec{u}+\vec{v}= diff --git a/gr2.tex b/gr2.tex index e55809f..2dac010 100644 --- a/gr2.tex +++ b/gr2.tex @@ -229,7 +229,8 @@ Another way to understand the vector sum is with the \index{vector!sum} Draw the parallelogram formed by the vectors$\vec{v}$and$\vec{w}$. -Then the sum$\vec{v}+\vec{w}$extends along the diagonal +Then the sum$\vec{v}+\vec{w}$\index{vector!sum}\index{addition of vectors} +extends along the diagonal to the far corner. \begin{center} \includegraphics{ch1.15} @@ -254,7 +255,7 @@ canonical representation ends at that point. \end{equation*} And, we do addition and scalar multiplication component-wise. -Having considered points, we next turn to lines. +Having considered points, we next turn to lines.\index{line} In$\Re^2$, the line through $$(1,2)$$ and $$(3,1)$$ is comprised of (the endpoints of) the vectors in this set. \begin{equation*} @@ -356,7 +357,7 @@$\set{\vec{p}+t_1\vec{v}_1+t_2\vec{v}_2+\cdots+t_k\vec{v}_k where $$\vec{v}_1,\ldots,\vec{v}_k\in\Re^n$$ and $k\leq n$ is a \definend{$$k$$-dimensional linear surface}\index{linear surface} -(or \definend{$$k$$-flat}\index{flat}). +(or \definend{$$k$$-flat}\index{flat, $k$-flat}). For example, in $\Re^4$ \begin{equation*} \set{\colvec[r]{2 \\ \pi \\ 3 \\ -0.5} diff --git a/gr3.tex b/gr3.tex index fd607d6..f098459 100644 --- a/gr3.tex +++ b/gr3.tex @@ -153,7 +153,7 @@ The answer is $x=5/2$ and $y=2$. %<*GaussJordanReduction> This extension of Gauss's Method is the \definend{Gauss-Jordan Method}\index{Gauss's Method!Gauss-Jordan Method} or -\definend{Gauss-Jordan reduction}.\index{linear equation!solution of!Gauss-Jordan}\index{Gauss-Jordan}\index{Gauss's Method!Gauss-Jordan} +\definend{Gauss-Jordan reduction}.\index{linear equation!solution of!Gauss-Jordan}\index{Gauss's Method!Gauss-Jordan} % % It goes past echelon form to a more refined, more specialized, % matrix form. diff --git a/homogeom.tex b/homogeom.tex index 1c27a2c..d1b0560 100644 --- a/homogeom.tex +++ b/homogeom.tex @@ -62,7 +62,8 @@ use the standard bases to represent it by a matrix $H$. Recall that $H$ factors into $H=PBQ$ where $P$ and $Q$ are nonsingular and $B$ is a partial-identity matrix. Recall also that nonsingular matrices -factor into elementary matrices\index{matrix!elementary reduction}\index{elementary!matrix} +factor into elementary +matrices\index{matrix!elementary reduction}\index{elementary reduction matrix} $PBQ=T_nT_{n-1}\cdots T_sBT_{s-1}\cdots T_1$, which are matrices that come from the identity $I$ after one Gaussian row operation, diff --git a/jc1.tex b/jc1.tex index c04ad08..b29f158 100644 --- a/jc1.tex +++ b/jc1.tex @@ -60,7 +60,8 @@ Consequently in this chapter we shall use complex numbers for our scalars, including entries in vectors and matrices. That is, we shift from studying vector spaces over the real numbers -to vector spaces over the complex numbers. +to vector spaces over the +complex numbers.\index{complex numbers!vector space over} Any real number is a complex number and in this chapter most of the examples use only real numbers but @@ -94,7 +95,7 @@ Consider a polynomial\index{polynomial} $p(x)=c_nx^n+\dots+c_1x+c_0$ with leading coefficient\index{polynomial!leading coefficient} $c_n\neq 0$ and $n\geq 1$. -The degree\index{polynomial!degree}\index{degree of polynomial} +The degree\index{polynomial!degree}\index{degree of a polynomial} of the polynomial is~$n$. If $n=0$ then $p$ is a constant polynomial\index{polynomial!constant}\index{constant polynomial} @@ -204,7 +205,7 @@ roots of $$ax^2+bx+c$$ are these has no real number roots). A polynomial that cannot be factored into two lower-degree polynomials with real number coefficients is said to be irreducible over the -reals.\index{irreducible}\index{polynomial!irreducible} +reals.\index{irreducible polynomial}\index{polynomial!irreducible} \begin{theorem} \label{th:CubicsAndHigherFactor} %<*th:CubicsAndHigherFactor> @@ -275,7 +276,7 @@ into the product of two first degree polynomials. \end{equation*} \end{example} -\begin{theorem}[Fundamental Theorem of Algebra] \label{th:FundThmAlg} +\begin{theorem}[Fundamental Theorem of Algebra] \label{th:FundThmAlg}\index{Fundamental Theorem!of Algebra} \hspace*{0em plus2em} %<*th:FundThmAlg> Polynomials with complex coefficients factor into linear @@ -351,8 +352,8 @@ For instance, we shall call this \dots, \colvec{0+0i \\ 0+0i \\ \vdots \\ 1+0i}} \end{equation*} -the \definend{standard basis\/}\index{basis!standard}% -\index{basis!standard over the complex numbers} +the \definend{standard basis}\index{standard basis}\index{basis!standard}% +\index{standard basis!complex number scalars} for $$\C^n$$ as a vector space over $\C$ and again denote it $$\stdbasis_n$$. diff --git a/jc2.tex b/jc2.tex index cfbf3a9..d4f76e0 100644 --- a/jc2.tex +++ b/jc2.tex @@ -2324,11 +2324,11 @@ where $$b\neq 0$$. \begin{definition} \label{df:CharacteristicPoly} %<*df:CharacteristicPoly> -The \definend{characteristic polynomial of a square matrix}\index{characteristic polynomial}% +The \definend{characteristic polynomial of a square matrix}\index{characteristic!polynomial}% \index{matrix!characteristic polynomial} $$T$$ is the determinant $$\deter{T-x I}$$ where $$x$$ is a variable. -The \definend{characteristic equation}\index{characteristic equation}% +The \definend{characteristic equation}\index{characteristic!equation}% \index{matrix!characteristic polynomial} is $\deter{T-xI}=0$. The \definend{characteristic polynomial of a transformation} @@ -3512,7 +3512,7 @@ Apply \nearbylemma{lm:DiagIffBasisOfEigens}. \cite{MathMag67p232} Show that if $$A$$ is an $$n$$ square matrix and each row (column) sums to $$c$$ then $$c$$ is a characteristic root of $$A$$. - (Characteristic root'' is a synonym for eigenvalue.)\index{characteristic root}\index{root!characteristic} + (Characteristic root'' is a synonym for eigenvalue.)\index{characteristic!root}\index{root!characteristic} \begin{answer} \answerasgiven % If the argument of the characteristic function of $$A$$ is set equal to diff --git a/jc3.tex b/jc3.tex index 8548b41..de3e87e 100644 --- a/jc3.tex +++ b/jc3.tex @@ -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}\index{index, of nilpotency} \end{definition} \begin{example} @@ -1844,7 +1844,7 @@ such that $$n(\vec{\beta}_1)=\vec{\beta}_2$$. that is, prove that $$t$$ restricted to the span has a range that is a subset of the span. We say that the span is a \definend{$$t$$-invariant} - subspace.\index{invariant!subspace} + subspace.\index{invariant subspace} \partsitem Prove that the restriction is nilpotent. \partsitem Prove that the $t$-string is linearly independent and so is a basis for its span. diff --git a/jc4.tex b/jc4.tex index ad26413..1174037 100644 --- a/jc4.tex +++ b/jc4.tex @@ -423,7 +423,7 @@ The total on the right is the zero matrix. We refer to that result by saying that a matrix or map -\definend{satisfies}\index{characteristic polynomial!satisfied by} +\definend{satisfies}\index{characteristic!polynomial!satisfied by} its characteristic polynomial. \begin{lemma} \label{le:tSatisImpMinPolyDivides} @@ -1992,7 +1992,7 @@ condition. \begin{definition} \label{def:invariant} Let $$\map{t}{V}{V}$$ be a transformation. A subspace $$M$$ is \definend{$t$ invariant}% -\index{invariant subspace!definition}\index{subspace!invariant} +\index{invariant subspace}\index{subspace!invariant} if whenever $$\vec{m}\in M$$ then $$t(\vec{m})\in M$$ (shorter: $$t(M)\subseteq M$$). \end{definition} diff --git a/map2.tex b/map2.tex index ea20bea..0d0836c 100644 --- a/map2.tex +++ b/map2.tex @@ -37,10 +37,10 @@ and scalar multiplication if $$\vec{v}\in V$$ and $$r\in\Re$$ then $$h(r\cdot\vec{v})=r\cdot h(\vec{v})$$ \end{center} -is a \definend{homomorphism}\index{homomorphism}% +is a \definend{homomorphism}\index{homomorphism}\index{linear map}% \index{function!structure preserving!\see{homomorphism}}% \index{vector space!homomorphism}\index{vector space!map} -or \definend{linear map}\index{linear map!see{homomorphism}}. +or \definend{linear map}\index{linear map|seealso{homomorphism}}. % \end{definition} @@ -282,7 +282,7 @@ let $B=\sequence{\vec{\beta}_1,\ldots,\vec{\beta}_n}$ be a basis for~$V$. A function defined on that basis $\map{f}{B}{W}$ -is \definend{extended linearly}\index{extended linearly}\index{function!extended linearly}\index{linear extension of a function} +is \definend{extended linearly}\index{extended, linearly}\index{function!extended linearly}\index{linear extension of a function} to a function $\map{\hat{f}}{V}{W}$ if for all $\vec{v}\in V$ such that $\vec{v}=c_1\vec{\beta}_1+\cdots+c_n\vec{\beta}_n$, @@ -320,7 +320,7 @@ like this one, using matrices. \begin{definition} \label{df:LinearTransformation} %<*df:LinearTransformation> A linear map from a space into itself $$\map{t}{V}{V}$$ is a -\definend{linear transformation}\index{linear transformation!see{transformation}}. +\definend{linear transformation}\index{linear transformation}\index{linear transformation|seealso{transformation}}. % \end{definition} @@ -398,7 +398,7 @@ from $$V$$ to $$W$$. %<*SpLinFcns> \noindent We denote the space of linear maps from $V$ to~$W$ by -$$\linmaps{V}{W}$$.\index{linear maps!space of} +$$\linmaps{V}{W}$$.\index{linear maps, vector space of} % \begin{proof} @@ -1712,7 +1712,7 @@ is a member of $S$. \begin{definition} \label{df:NullSpace} %<*df:NullSpace> The \definend{null space}\index{homomorphism!null space}\index{null space} -or \definend{kernel}\index{kernel} of a linear map +or \definend{kernel}\index{kernel, of linear map} of a linear map $$\map{h}{V}{W}$$ is the inverse image of $\zero_W$. \begin{equation*} \nullspace{h}=h^{-1}(\zero_W)=\set{\vec{v}\in V\suchthat h(\vec{v})=\zero_W} diff --git a/map4.tex b/map4.tex index 020a856..d1c8257 100644 --- a/map4.tex +++ b/map4.tex @@ -2862,7 +2862,7 @@ perform the combination operation $$-2\rho_2+\rho_3$$. \begin{definition} \label{df:ElementaryReductionMatrices} %<*df:ElementaryReductionMatrices> The \definend{elementary reduction matrices}% -\index{matrix!elementary reduction}\index{elementary!matrix} +\index{matrix!elementary reduction}\index{elementary reduction matrix} result from applying a one Gaussian operation to an identity matrix. \begin{enumerate} \item $$I\grstep{k\rho_i}M_i(k)$$ for $$k\neq 0$$ @@ -2875,7 +2875,7 @@ result from applying a one Gaussian operation to an identity matrix. \begin{lemma} \label{GrByMatMult} %<*lm:GrByMatMult> -Matrix multiplication can do Gaussian reduction. +Matrix multiplication can do Gaussian reduction.\index{elementary reduction operations!by matrix multiplication}\index{elementary row operations!by matrix multiplication}\index{Gauss's Method!by matrix multiplication} \begin{enumerate} \item If $$H\grstep{k\rho_i}G$$ then $$M_i(k)H=G$$. \item If $$H\grstep{\rho_i\leftrightarrow\rho_j}G$$ diff --git a/markov.tex b/markov.tex index 2e8567d..52d8841 100644 --- a/markov.tex +++ b/markov.tex @@ -148,7 +148,7 @@ whose entries are nonnegative reals and whose columns sum to $1$. A characteristic feature of a Markov chain model is that it is -\definend{historyless}\index{historyless}% +\definend{historyless}\index{historyless process}% \index{Markov chain!historyless} in that the next state depends only on the current state, not on any prior ones. diff --git a/ppivot.tex b/ppivot.tex index 49c384b..6d6b722 100644 --- a/ppivot.tex +++ b/ppivot.tex @@ -84,7 +84,7 @@ The solution changes radically depending on the ninth digit, which explains why an eight-place computer has trouble. A problem that is very sensitive to inaccuracy or uncertainties in -the input values is \definend{ill-conditioned}.\index{ill-conditioned} +the input values is \definend{ill-conditioned}.\index{ill-conditioned problem} The above example gives one way in which a system can be difficult to solve on a computer. diff --git a/projplane.tex b/projplane.tex index 1241879..de141ee 100644 --- a/projplane.tex +++ b/projplane.tex @@ -327,11 +327,11 @@ Thus we can think of projective space as consisting of the Euclidean plane with some extra points adjoined \Dash the Euclidean plane is embedded in the projective plane. The extra points in projective space, the equatorial points, -are called \definend{ideal points}\index{ideal point}% +are called \definend{ideal points}\index{ideal!point}% \index{projective plane!ideal point} or \definend{points at infinity}\index{point!at infinity} and the equator is called the -\definend{ideal line}\index{ideal line}% +\definend{ideal line}\index{ideal!line}% \index{projective plane!ideal line} or \definend{line at infinity}\index{line at infinity} (it is not a Euclidean line, it is a projective line). diff --git a/vs1.tex b/vs1.tex index 30c80f5..b35ee78 100644 --- a/vs1.tex +++ b/vs1.tex @@ -92,8 +92,7 @@ operations +' and `$$\cdot$$' subject to these conditions. %<*df:VectorSpace1> Where $$\vec{v},\vec{w}\in V$$, -(1)~their \definend{vector sum}\index{vector!sum}\index{sum!vector}% - \index{addition!vector} +(1)~their \definend{vector sum}\index{vector!sum}\index{sum!vector}\index{addition of vectors} $$\vec{v}+\vec{w}$$ is an element of $$V\/$$. If $$\vec{u},\vec{v},\vec{w}\in V$$ then (2)~$$\vec{v}+\vec{w}=\vec{w}+\vec{v}$$ and diff --git a/vs3.tex b/vs3.tex index acff11b..a41e528 100644 --- a/vs3.tex +++ b/vs3.tex @@ -2441,9 +2441,9 @@ not directly involving row vectors. \begin{definition} \label{df:ColumnSpace} %<*df:ColumnSpace> -The \definend{column space\/}\index{column space}\index{matrix!column space} +The \definend{column space}\index{column!space}\index{matrix!column space} of a matrix is the span of the set of its columns. -The \definend{column rank\/}\index{column!rank}\index{rank!column} +The \definend{column rank}\index{column!rank}\index{rank!column} is the dimension of the column space, the number of linearly independent columns. % @@ -3475,7 +3475,7 @@ Where there are $$r$$ independent equations, the general solution involves \end{answer} \item Show that the transpose operation is - \definend{linear}:\index{linear!transpose operation} + linear:\index{transpose!is linear} \begin{equation*} \trans{(rA+sB)} = r\trans{A}+s\trans{B} \end{equation*} @@ -3708,7 +3708,7 @@ Where there are $$r$$ independent equations, the general solution involves \definend{full row rank}\index{full row rank}\index{row rank!full} if its row rank is $$m$$, and it has - \definend{full column rank}\index{full column rank}\index{column rank!full} + \definend{full column rank}\index{full column rank}\index{column!rank!full} if its column rank is $$n$$. \begin{exparts} \partsitem Show that @@ -4096,7 +4096,7 @@ has one and only one solution for any $x,y,z\in\Re$. % each vector decomposes uniquely into a sum of vectors from the parts. \begin{definition} -The \definend{concatenation}\index{concatenation}% +The \definend{concatenation}\index{concatenation of sequences}% \index{sequence!concatenation} of the sequences \$ -- 2.22.0