Update linear algebra index

source/linear-algebra/source/01-LE/01.ptx

@@ -127,7 +127,7 @@ When order is irrelevant, we will use set notation:
     <definition>
         <statement>
             <p>
-A <term>Euclidean vector</term><idx>Euclidean vector</idx><idx><h>vector</h><h>Euclidean</h></idx> is an ordered
+A <term>Euclidean vector</term><idx><h>Euclidean</h><h>vector</h></idx> is an ordered
 list of real numbers
 <me>
   \left[\begin{array}{c}
@@ -266,7 +266,7 @@ a_1x_1+a_2x_2+\dots+a_nx_n=b
 </me>.
             </p>
             <p>
-                A <term>solution</term><idx><h>linear equation</h><h>solution</h></idx> for a linear equation is a Euclidean vector
+                A <term>solution</term><idx><h>solution</h><h>of a linear equation</h></idx> for a linear equation is a Euclidean vector
 <me>
   \left[\begin{array}{c}
     s_1 \\
@@ -415,9 +415,9 @@ is equivalent to the system of equations
     <definition>
         <statement>
             <p>
-A linear system is <term>consistent</term><idx><h>linear system</h><h>consistent</h></idx> if its solution set
+A linear system is <term>consistent</term><idx><h>consistent linear system</h></idx> if its solution set
   is non-empty (that is, there exists a solution for the
-system). Otherwise it is <term>inconsistent</term>.<idx><h>linear system</h><h>inconsistent</h></idx>
+system). Otherwise it is <term>inconsistent</term>.<idx><h>inconsistent linear system</h></idx>
             </p>
         </statement>
     </definition>
@@ -782,7 +782,7 @@ Then use these to describe the solution set
             Sometimes, we will find it useful to refer only to the coefficients of the linear system
             (and ignore its constant terms).
             We call the <m>m\times n</m> array consisting of these coefficients 
-            a <term>coefficient matrix</term><idx>coeefficient matrix</idx>.
+            a <term>coefficient matrix</term><idx>coefficient matrix</idx>.
             <me>
                 \left[\begin{array}{cccc}
                   a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n}\\

source/linear-algebra/source/01-LE/02.ptx

@@ -556,7 +556,7 @@ matrix to the next:
 <definition>
     <statement>
         <p>
-A matrix is in <term>reduced row echelon form</term> (<term>RREF</term>)<idx>Reduced row echelon form</idx> if
+A matrix is in <term>reduced row echelon form</term> (<term>RREF</term>)<idx>reduced row echelon form</idx> if
 <ol>
 <li> The leftmost nonzero term of each row is 1.
       We call these terms <term>pivots</term>.<idx>pivot</idx>

source/linear-algebra/source/01-LE/03.ptx

@@ -467,7 +467,7 @@ different solutions. We'll learn how to find such solution sets in
         <title>Mathematical Writing Explorations</title>
                 <exploration>
             <statement>
-                <p> A system of equations with all constants equal to 0 is called <term> homogeneous</term>. These are addressed in detail in section <xref ref="EV7"></xref>
+                <p> A system of equations with all constants equal to 0 is called <term> homogeneous</term><idx>homogeneous</idx>. These are addressed in detail in section <xref ref="EV7"></xref>
                 <ul>
                     <li> Choose three systems of equations from this chapter that you have already solved. Replace the constants with 0 to make the systems homogeneous. Solve the homogeneous systems and make a conjecture about the relationship between the earlier solutions you found and the associated homogeneous systems.
                     </li>

source/linear-algebra/source/02-EV/01.ptx

@@ -68,7 +68,7 @@ we refer to this real number as a <term>scalar</term>.
 <definition xml:id="EV1-definition-linear-combo">
     <statement>
         <p>
-  A <term>linear combination</term> <idx> linear combination</idx>of a set of vectors
+  A <term>linear combination</term> <idx>linear combination</idx>of a set of vectors
   <m>\{\vec v_1,\vec v_2,\dots,\vec v_n\}</m> is given by
   <m>c_1\vec v_1+c_2\vec v_2+\dots+c_n\vec v_n</m> for any choice of
   scalar multiples <m>c_1,c_2,\dots,c_n</m>.

source/linear-algebra/source/02-EV/03.ptx

@@ -317,12 +317,12 @@
         </li>
         <li>
             <p>
-            the subset is <term>closed under addition</term><idx><h>vector space</h></idx><idx>closed under addition</idx>: for any <m>\vec{u},\vec{v} \in W</m>, the sum <m>\vec{u}+\vec{v}</m> is also in <m>W</m>.
+            the subset is <term>closed under addition</term><idx><h>closed</h><h>under addition</h></idx>: for any <m>\vec{u},\vec{v} \in W</m>, the sum <m>\vec{u}+\vec{v}</m> is also in <m>W</m>.
             </p>
         </li>
         <li>
             <p>
-            the subset is <term>closed under scalar multiplication</term><idx><h>vector space</h></idx><idx>closed under scalar multiplication</idx>: for any <m>\vec{u} \in W</m> and scalar <m>c \in \IR</m>, the product <m>c\vec{u}</m> is also in <m>W</m>.
+            the subset is <term>closed under scalar multiplication</term><idx><h>closed</h><h>under scalar multiplication</h></idx>: for any <m>\vec{u} \in W</m> and scalar <m>c \in \IR</m>, the product <m>c\vec{u}</m> is also in <m>W</m>.
             </p>
         </li>
         </ul>

source/linear-algebra/source/02-EV/05.ptx

@@ -244,7 +244,7 @@ that equals the vector
 <definition>
     <statement>
         <p>
-  The <term>standard basis</term><idx><h>basis</h></idx><idx>standard</idx> of <m>\IR^n</m> is the set <m>\{\vec{e}_1, \ldots, \vec{e}_n\}</m> where
+  The <term>standard basis</term><idx><h>standard</h><h>basis</h></idx> of <m>\IR^n</m> is the set <m>\{\vec{e}_1, \ldots, \vec{e}_n\}</m> where
             <md>
                 <mrow>
   \vec{e}_1 &amp;= \left[\begin{array}{c}1 \\ 0 \\ 0 \\ \vdots \\ 0 \\  0 \end{array}\right] &amp;

source/linear-algebra/source/02-EV/06.ptx

@@ -128,7 +128,7 @@ is now independent.
 <definition>
     <statement>
         <p>
-  Let <m>W</m> be a subspace of a vector space. A <term>basis</term> for
+  Let <m>W</m> be a subspace of a vector space. A <term>basis</term><idx>basis</idx> for
   <m>W</m> is a linearly independent set of vectors that spans <m>W</m>
   (but not necessarily the entire vector space).
         </p>

source/linear-algebra/source/02-EV/07.ptx

@@ -458,7 +458,7 @@ shape of the data that gets projected onto this single point of the screen?
 <subsection>    
     <title>Mathematical Writing Explorations</title>
             <exploration xml:id="non-singular">
-               <p> An <m>n \times n</m> matrix <m>M</m> is <term>non-singular</term><idx>non-singular</idx> if the associated homogeneous system with coefficient matrix <m>M</m> is consistent with one solution. Assume the matrices in the writing explorations in this section are all non-singular.
+               <p> An <m>n \times n</m> matrix <m>M</m> is <term>non-singular</term><idx>non-singular matrix</idx> if the associated homogeneous system with coefficient matrix <m>M</m> is consistent with one solution. Assume the matrices in the writing explorations in this section are all non-singular.
 
 <ul>
 <li>Prove that the reduced row echelon form of <m>M</m> is the identity matrix.

source/linear-algebra/source/03-AT/01.ptx

@@ -79,8 +79,8 @@ can be applied before or after the transformation without affecting the result.
     <statement>
         <p>
 Given a linear transformation <m>T:V\to W</m>,
-<m>V</m> is called the <term>domain</term> of <m>T</m> and
-<m>W</m> is called the <term>co-domain</term> of <m>T</m>.
+<m>V</m> is called the <term>domain</term><idx><h>domain</h></idx> of <m>T</m> and
+<m>W</m> is called the <term>co-domain</term><idx><h>co-domain</h></idx> of <m>T</m>.
         </p>
     <figure>
     <caption>A linear transformation with a domain of <m>\IR^3</m> and a co-domain of <m>\IR^2</m></caption>

source/linear-algebra/source/03-AT/02.ptx

@@ -298,7 +298,7 @@ Therefore any linear transformation <m>T:V \rightarrow W</m> can be defined
 by just describing the values of <m>T(\vec b_i)</m>.
         </p>
         <p>
-Put another way, the images of the basis vectors completely <term>determine</term><idx>determine</idx> the transformation <m>T</m>.
+Put another way, the images of the basis vectors completely determine the transformation <m>T</m>.
         </p>
     </statement>
 </fact>

source/linear-algebra/source/03-AT/03.ptx

@@ -84,7 +84,7 @@ the set of all vectors that transform into <m>\vec 0</m>?
     <statement>
         <p>
 Let <m>T: V \rightarrow W</m> be a linear transformation, and let <m>\vec{z}</m> be the additive
-identity (the <q>zero vector</q>) of <m>W</m>.  The <term>kernel</term><idx>kernel</idx> of <m>T</m>
+identity (the <q>zero vector</q>) of <m>W</m>.  The <term>kernel</term><idx>kernel</idx>of <m>T</m>
 (also known as the <term>null space</term><idx>null space</idx> of <m>T</m>)
 is an important subspace of <m>V</m> defined by
 <me>
@@ -656,15 +656,15 @@ Which of the following is equal to the dimension of the image of <m>T</m>?
 
 <observation>
         <p>
-Combining these with the observation that the number of columns is the dimension of the domain of <m>T</m>, we have the <term>rank-nullity theorem</term>:
+Combining these with the observation that the number of columns is the dimension of the domain of <m>T</m>, we have the <term>rank-nullity theorem</term><idx>rank-nullity theorem</idx>:
         </p>
         <blockquote>
         <p>
 The dimension of the domain of <m>T</m> equals <m>\dim(\ker T)+\dim(\Im T)</m>.
         </p>
         </blockquote>
         <p>
-The dimension of the image is called the <term>rank</term> of <m>T</m> (or <m>A</m>) and the dimension of the kernel is called the <term>nullity</term>. 
+The dimension of the image is called the <term>rank</term><idx><h>rank</h><h>of a linear transformation</h></idx> of <m>T</m> (or <m>A</m>) and the dimension of the kernel is called the <term>nullity</term><idx>nullity</idx>. 
         </p>
 </observation>
 

source/linear-algebra/source/03-AT/04.ptx

@@ -39,7 +39,7 @@
     <statement>
         <p>
 Let <m>T: V \rightarrow W</m> be a linear transformation.
-<m>T</m> is called <term>injective</term> or <term>one-to-one</term> if <m>T</m> does not map two
+<m>T</m> is called <term>injective</term><idx>injective</idx> or <term>one-to-one</term><idx><seealso>injective</seealso></idx> if <m>T</m> does not map two
 distinct vectors to the same place.  More precisely, <m>T</m> is injective if
 <m>T(\vec{v}) \neq T(\vec{w})</m> whenever <m>\vec{v} \neq \vec{w}</m>.
         </p>
@@ -208,7 +208,7 @@ Is <m>T</m> injective?
     <statement>
         <p>
 Let <m>T: V \rightarrow W</m> be a linear transformation.
-<m>T</m> is called <term>surjective</term> or <term>onto</term> if every element of <m>W</m> is mapped to by an element of <m>V</m>.  More precisely, for every <m>\vec{w} \in W</m>, there is some <m>\vec{v} \in V</m> with <m>T(\vec{v})=\vec{w}</m>.
+<m>T</m> is called <term>surjective</term><idx>surjective</idx> or <term>onto</term><idx><seealso>surjective</seealso></idx> if every element of <m>W</m> is mapped to by an element of <m>V</m>.  More precisely, for every <m>\vec{w} \in W</m>, there is some <m>\vec{v} \in V</m> with <m>T(\vec{v})=\vec{w}</m>.
         </p>
     <figure>
 		<caption>A surjective transformation and a non-surjective transformation</caption>

source/linear-algebra/source/03-AT/06.ptx

@@ -555,7 +555,7 @@ as a linear combination of polynomials from the set
             <exploration>
                 <statement>
               <p>
-                    Given a matrix <m>M</m> the <term>rank</term><idx>rank</idx> of a matrix is the dimension of the column space.
+                    Given a matrix <m>M</m> the <term>rank</term><idx><h>rank</h><h>of a matrix</h></idx> of a matrix is the dimension of the column space.
 
 
 

source/linear-algebra/source/04-MX/01.ptx

@@ -147,7 +147,7 @@ What are the domain and codomain of the composition map <m>S \circ T</m>?
 <definition>
     <statement>
     <p>
-We define the <term>product</term> <m>AB</m> of a <m>m \times n</m> matrix <m>A</m> and a
+We define the <term>product</term><idx><h>matrix</h><h>product</h></idx> <m>AB</m> of a <m>m \times n</m> matrix <m>A</m> and a
 <m>n \times k</m>
 matrix <m>B</m> to be the <m>m \times k</m> standard matrix of the composition map of the
 two corresponding linear functions.

source/linear-algebra/source/04-MX/02.ptx

@@ -108,7 +108,7 @@ Let <m>T: \IR^n \rightarrow \IR^n</m> be a linear bijection with standard matrix
         </p>
         <p>
 By item (B) from <xref ref="MX2-card-sort"/>
-we may define an <term>inverse map</term><idx>inverse map</idx>
+we may define an <term>inverse map</term><idx>inverse map</idx><idx><h>invertible</h><h>linear transformation</h></idx>
 <m>T^{-1} : \IR^n \rightarrow \IR^n</m> 
 that defines <m>T^{-1}(\vec b)</m> as the unique solution <m>\vec x</m> satisfying
 <m>T(\vec x)=\vec b</m>, that is, <m>T(T^{-1}(\vec b))=\vec b</m>.
@@ -118,7 +118,7 @@ Furthermore, let
 <me>A^{-1}=[T^{-1}(\vec e_1)\hspace{1em}\cdots\hspace{1em}T^{-1}(\vec e_n)]</me>
 be the standard matrix for <m>T^{-1}</m>. We call <m>A^{-1}</m> the
 <term>inverse matrix</term><idx>inverse matrix</idx> of <m>A</m>,
-and we also say that <m>A</m> is an <term>invertible</term><idx>invertible</idx>
+and we also say that <m>A</m> is an <term>invertible</term><idx><h>invertible</h><h>matrix</h></idx>
 matrix.
         </p>
     </statement>

source/linear-algebra/source/05-GT/01.ptx

@@ -252,7 +252,7 @@ begins with area <m>1</m>).
 <remark>
     <statement>
     <p>
-We will define the <term>determinant</term> of a square matrix <m>B</m>,
+We will define the <term>determinant</term><idx>determinant</idx> of a square matrix <m>B</m>,
 or <m>\det(B)</m> for short, to be the factor by which <m>B</m> scales areas.
 In order to figure out how to compute it, we first figure out the properties it must satisfy.
     </p>
@@ -286,7 +286,7 @@ In order to figure out how to compute it, we first figure out the properties it
     The images from <xref ref="fig-GT1-image-unit-square-transform2"/> and <xref ref="fig-GT1-image-scale-not-rotate"/> are shown side by side.
   </description>
 	</image>
-	<caption>The linear transformation <m>B</m> scaling areas by a constant factor, which we call the <term>determinant</term></caption>
+	<caption>The linear transformation <m>B</m> scaling areas by a constant factor, which we call the <term>determinant</term><idx>determinant</idx></caption>
 </figure>
     </statement>
 </remark>
@@ -429,7 +429,7 @@ the areas on the left may be partitioned and reorganized into the area on the ri
 <definition>
     <statement>
     <p>
-The <term>determinant</term> is the unique function
+The <term>determinant</term><idx>determinant</idx> is the unique function
 <m>\det:M_{n,n}\to\IR</m> satisfying these  properties:
 <ol>
 	<li><m>\det(I)=1</m></li>

source/linear-algebra/source/05-GT/03.ptx

@@ -173,7 +173,7 @@ is a vector <m>\vec{x} \in \IR^n</m> such that <m>A\vec{x}</m> is parallel to <m
 </figure>
 <p>
 In other words, <m>A\vec{x}=\lambda \vec{x}</m> for some scalar <m>\lambda</m>. 
-If <m>\vec x\not=\vec 0</m>, then we say <m>\vec x</m> is a <term>nontrivial eigenvector</term><idx><h>eigenvector</h></idx><idx>nontrivial</idx>
+If <m>\vec x\not=\vec 0</m>, then we say <m>\vec x</m> is a <term>nontrivial eigenvector</term><idx><h>nontrivial</h><h>eigenvector</h></idx>
 and we call this <m>\lambda</m> an <term>eigenvalue</term><idx>eigenvalue</idx> of <m>A</m>.
     </p>
     </statement>
@@ -272,7 +272,7 @@ And what else?
     <statement>
         <p>
 The expression <m>\det(A-\lambda I)</m> is called the
-<term>characteristic polynomial</term> of <m>A</m>.
+<term>characteristic polynomial</term><idx>characteristic polynomial</idx> of <m>A</m>.
         </p>
         <p>
 For example, when