Chapter 3 tweaks (#1028)

* Typeset with dfrac

* Section 3

* Section 4

* Section 6

Author: siwelwerd

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

@@ -752,7 +752,7 @@ T\left(\left[\begin{array}{c} -5 \\ 0 \\ -3 \\ -2 \end{array}\right]\right)
 
 <li>A <term>rotation</term> is given by <m>\vec{v} \mapsto \left[\begin{array}{c} \cos(\theta)x - \sin(\theta)y\\ \cos(\theta)y + \sin(\theta)x\end{array}\right].</m></li>
 
-<li>A <term>reflection</term> of <m>\vec{v}</m> over a line <m>l</m> can be found by first finding a vector <m>\vec{l} = \left[\begin{array}{c} l_x\\l_y\end{array}\right]</m> along <m>l</m>, then <m>\vec{v} \mapsto 2\frac{\vec{l}\cdot\vec{v}}{\vec{l}\cdot\vec{l}}\vec{l} - \vec{v}.</m></li>
+<li>A <term>reflection</term> of <m>\vec{v}</m> over a line <m>l</m> can be found by first finding a vector <m>\vec{l} = \left[\begin{array}{c} l_x\\l_y\end{array}\right]</m> along <m>l</m>, then <m>\vec{v} \mapsto 2 \left(\dfrac{\vec{l}\cdot\vec{v}}{\vec{l}\cdot\vec{l}}\right) \vec{l} - \vec{v}.</m></li>
 </ul>
 Represent each of the following transformations with respect to the standard basis in <m>\mathbb{R}^2</m>.
 <ul>

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>
@@ -107,7 +107,7 @@ is an important subspace of <m>V</m> defined by
       \draw (-2,0) -- (2,0);
       \draw (0,-2) -- (0,2);
       \fill[blue] (0,0) circle (0.2)
-            node[anchor=south east] {\(\vec{0}\)};
+            node[anchor=south east] {\(\vec{z}\)};
     \end{scope}
   \end{tikzpicture}
 					</latex-image>

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

@@ -1000,7 +1000,7 @@ Which of the following must be true?
         </li>
         <li>
           <p>
-            The system of equations given by <m>[A|\vec{0}]</m> has a unique solution. 
+            The system of equations given by <m>[A\,|\,\vec{0}]</m> has a unique solution. 
           </p>
         </li>
       </ol>
@@ -1046,7 +1046,7 @@ Which of the following must be true?
         </li>
         <li>
           <p>
-            The system of equations given by <m>[A|\vec{b}]</m> is always consistent. 
+            The system of equations given by <m>[A\,|\,\vec{b}]</m> is always consistent. 
           </p>
         </li>
       </ol>
@@ -1108,7 +1108,7 @@ Which of the following must be true?
  <exploration><statement>
      <p>Start with an <m>n</m>-dimensional vector space <m>V</m>. We can define the <term>dual</term> of <m>V</m>, denoted <m>V^*</m>, by 
 <me>V^* = \{h:V \rightarrow \mathbb{R}: h \mbox{ is linear}\}.</me>
-Prove that <m>V</m> is isomorphic to<m>V^*</m>. Here are some things to think about as you work through this.
+Prove that <m>V</m> is isomorphic to <m>V^*</m>. Here are some things to think about as you work through this.
 <ul>
 <li>Start by assuming you have a basis for <m>V</m>. How many basis vectors should you have?</li>
 <li>For each basis vector in <m>V</m>, define a function that returns 1 if it's given that basis vector, and returns 0 if it's given any other basis vector. For example, if <m>\vec{b_i}</m> and <m>\vec{b_j}</m> are each members of the basis for <m>V</m>, and you'll need a function <m>f_i:V \rightarrow \{0,1\}</m>, where <m>f_i(b_i) = 1</m> and <m>f_i(b_j)= 0</m> for all <m> j \neq i</m>.</li>

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

@@ -312,15 +312,15 @@ Consider the matrix space <m>M_{2,2}=\left\{\left[\begin{array}{cc}
 <activity>
     <introduction>
         <p>
-Consider polynomial space <m>\P^4=\left\{a+by+cy^2+dy^3+ey^4\middle| a,b,c,d,e\in\IR\right\}</m> and the following set:
+Consider polynomial space <m>\P_4=\left\{a+by+cy^2+dy^3+ey^4\middle| a,b,c,d,e\in\IR\right\}</m> and the following set:
 <me>
 S=\setList{1,y,y^2,y^3,y^4}.
 </me>
         </p>
     </introduction>
     <task>
         <statement>
-            <p> Does the set <m>S</m> span <m>\P^4</m>?
+            <p> Does the set <m>S</m> span <m>\P_4</m>?
                 <ol marker="A.">
                     <li>
                         <p>
@@ -339,7 +339,7 @@ S=\setList{1,y,y^2,y^3,y^4}.
                     </li>
                     <li>
                         <p>
-                            Yes; every polynomial in <m>\P^4</m> is a linear combination of the polynomials in <m>S</m>.
+                            Yes; every polynomial in <m>\P_4</m> is a linear combination of the polynomials in <m>S</m>.
                         </p>
                     </li>
                 </ol>
@@ -384,15 +384,15 @@ S=\setList{1,y,y^2,y^3,y^4}.
                 </me>
                 <ol marker="A." >
                     <li><m>S</m> is linearly independent</li>
-                    <li><m>S</m> spans <m>\P^4</m></li>
-                    <li><m>S</m> is a basis of <m>\P^4</m></li>
+                    <li><m>S</m> spans <m>\P_4</m></li>
+                    <li><m>S</m> is a basis of <m>\P_4</m></li>
                 </ol>
             </p>
         </statement>
     </task>
     <task>
         <statement>
-            <p> What is the dimension of <m>\P^4</m>?
+            <p> What is the dimension of <m>\P_4</m>?
                 <ol marker="A." cols="4">
                     <li>2</li>
                     <li>3</li>
@@ -405,7 +405,7 @@ S=\setList{1,y,y^2,y^3,y^4}.
     <task>
         <statement>
             <p>
-                Which Euclidean space is <m>\P^4</m> isomorphic to?
+                Which Euclidean space is <m>\P_4</m> isomorphic to?
                 <ol marker="A." cols="4">
                     <li><m>\IR^2</m></li>
                     <li><m>\IR^3</m></li>
@@ -418,7 +418,7 @@ S=\setList{1,y,y^2,y^3,y^4}.
     <task>
         <statement>
             <p>
-                Describe an isomorphism <m>T:\P^4\to\IR^{\unknown}</m>:
+                Describe an isomorphism <m>T:\P_4\to\IR^{\unknown}</m>:
                 <me>
                     T\left(a+by+cy^2+dy^3+ey^4\right)=\left[\begin{array}{c}
                         \unknown\\\\\vdots\\\\\unknown