Fix typo in rank-thm.xml

src/invertible-matrix-thm.xml

@@ -201,7 +201,7 @@ the license is included in gfdl.xml.
    <example>
      <statement>
        <p>
-         Suppose that <m>A</m> is an <m>n\times n</m> matrix such that <m>Ax=b</m> is inconsistent some vector <m>b</m>.  Show that <m>Ax=b</m> has infinitely many solutions for some (other) vector <m>b</m>.
+         Suppose that <m>A</m> is an <m>n\times n</m> matrix such that <m>Ax=b</m> is inconsistent for some vector <m>b</m>.  Show that <m>Ax=b</m> has infinitely many solutions for some (other) vector <m>b</m>.
        </p>
      </statement>
      <solution>

src/linear-trans.xml

@@ -224,7 +224,7 @@ the license is included in gfdl.xml.
   \end{scope}
 \end{tikzpicture}
           </latex-code>
-          For the second property, <m>cT(u)</m> is the vector obtained by rotating <m>u</m> by the angle <m>\theta</m>, then changing its length by a factor of <m>c</m> (reversing direction of <m>c&lt;0</m>.  On the other hand, <m>T(cu)</m> first changes the length of <m>c</m>, then rotates.  But it does not matter in which order we do these two operations.
+          For the second property, <m>cT(u)</m> is the vector obtained by rotating <m>u</m> by the angle <m>\theta</m>, then changing its length by a factor of <m>c</m> (reversing direction if <m>c&lt;0</m>).  On the other hand, <m>T(cu)</m> first changes the length of <m>c</m>, then rotates.  But it does not matter in which order we do these two operations.
           <latex-code mode="bare">
             \usetikzlibrary{angles}
           </latex-code>
@@ -439,7 +439,7 @@ the license is included in gfdl.xml.
       <idx><h>Standard coordinate vectors</h><h>and matrix columns</h></idx>
       <idx><h>Matrix-vector product</h><h>with standard coordinate vectors</h></idx>
       <p>
-        If <m>A</m> is an <m>m\times n</m> matrix with columns <m>v_1,v_2,\ldots,v_m</m>, then <m>\color{red}Ae_i = v_i</m> for each <m>i=1,2,\ldots,n</m>:
+        If <m>A</m> is an <m>m\times n</m> matrix with columns <m>v_1,v_2,\ldots,v_n</m>, then <m>\color{red}Ae_i = v_i</m> for each <m>i=1,2,\ldots,n</m>:
         <me>
           \mat{| | ,, |; v_1 v_2 \cdots, v_n; | | ,, |}e_i = v_i.
         </me>

src/matrix-inv.xml

@@ -520,9 +520,9 @@ $\hmat{\r1 0 3 0 1 0; 2 3 2 1 0 0; 2 2 3 0 0 1}$} \\[2mm]
         In other words, taking the cube root undoes the transformation that takes a number to its cube.
       </p>
       <p>
-        Define <m>f\colon\R\to\R</m> by <m>f(x) = x^2</m>.  This is <em>not</em> an invertible function.  Indeed, we have <m>f(2) = 2 = f(-2)</m>, so there is no way to undo <m>f</m>: the inverse transformation would not know if it should send <m>2</m> to <m>2</m> or <m>-2</m>.  More formally, if <m>g\colon\R\to\R</m> satisfies <m>g(f(x)) = x</m>, then
-        <me>2 = g(f(2)) = g(2) \sptxt{and} -2 = g(f(-2)) = g(2),</me>
-        which is impossible: <m>g(2)</m> is a number, so it cannot be equal to <m>2</m> and <m>-2</m> at the same time.
+        Define <m>f\colon\R\to\R</m> by <m>f(x) = x^2</m>.  This is <em>not</em> an invertible function.  Indeed, we have <m>f(2) = 4 = f(-2)</m>, so there is no way to undo <m>f</m>: the inverse transformation would not know if it should send <m>4</m> to <m>2</m> or <m>-2</m>.  More formally, if <m>g\colon\R\to\R</m> satisfies <m>g(f(x)) = x</m>, then
+        <me>2 = g(f(2)) = g(4) \sptxt{and} -2 = g(f(-2)) = g(4),</me>
+        which is impossible: <m>g(4)</m> is a number, so it cannot be equal to <m>2</m> and <m>-2</m> at the same time.
       </p>
       <p>
         Define <m>f\colon\R\to\R</m> by <m>f(x) = e^x</m>.  This is <em>not</em> an invertible function.  Indeed, if there were a function <m>g\colon\R\to\R</m> such that <m>f\circ g = \Id_\R</m>, then we would have

src/matrix-trans.xml

@@ -902,7 +902,7 @@ the license is included in gfdl.xml.
   </example>
 
   <example xml:id="matrix-trans-questions">
-    <title>Questions about a [matrix] transformation</title>
+    <title>Questions about a matrix transformation</title>
     <statement>
       <p>
         Let
@@ -979,7 +979,7 @@ the license is included in gfdl.xml.
   </example>
 
   <example xml:id="matrix-trans-questions-nonlinear">
-    <title>Questions about a [non-matrix] transformation</title>
+    <title>Questions about a non-matrix transformation</title>
     <statement>
       <p>
         Define a transformation <m>T\colon\R^2\to\R^3</m> by the formula

src/rank-thm.xml

@@ -243,7 +243,7 @@ the license is included in gfdl.xml.
 	</p>
 
 	<p>
-	 The rank theorem theorem is really the culmination of this chapter, as it gives a strong relationship between the null space of a matrix (the solution set of <m>Ax=0</m>) with the column space (the set of vectors <m>b</m> making <m>Ax=b</m> consistent), our two primary objects of interest.  The more freedom we have in choosing <m>x</m> the less freedom we have in choosing <m>b</m> and vice versa.  
+	 The rank theorem is really the culmination of this chapter, as it gives a strong relationship between the null space of a matrix (the solution set of <m>Ax=0</m>) with the column space (the set of vectors <m>b</m> making <m>Ax=b</m> consistent), our two primary objects of interest.  The more freedom we have in choosing <m>x</m> the less freedom we have in choosing <m>b</m> and vice versa.  
     </p>
 
     <example>