Finished all chapters and definitions. I need to add subsections and see if there's any theorems or definitions in the appendicies that are worth adding to this as well.

chapter-4/determinants-of-order-2.tex

@@ -3,46 +3,46 @@
 \begin{definition}
 	\hfill\\
 	If
-	
+
 	\[A = \begin{pmatrix}
-		a & b \\
-		c & d
-	\end{pmatrix}\]
-	 is a $2 \times 2$ matrix with entries from a field $\F$, then we define the \textbf{determinant} of $A$, denoted $\det(A)$ or $|A|$, to be the scalar $ad-bc$.
+			a & b \\
+			c & d
+		\end{pmatrix}\]
+	is a $2 \times 2$ matrix with entries from a field $\F$, then we define the \textbf{determinant} of $A$, denoted $\det(A)$ or $|A|$, to be the scalar $ad-bc$.
 \end{definition}
 
 \begin{theorem}
 	\hfill\\
 	The function $\det: M_{2 \times 2}(\F) \to \F$ is a linear function of each row of a $2 \times 2$ matrix when the other row is held fixed. That is, if $u$, $v$ and $w$ are in $\F^2$ and $k$ is a scalar, then
-	
+
 	\[\det \begin{pmatrix}
-		u + kv \\
-		w
-	\end{pmatrix} = \det\begin{pmatrix}
-	u \\ w
-	\end{pmatrix} + k\det\begin{pmatrix}
-	v \\ w
-	\end{pmatrix}\]
-	
+			u + kv \\
+			w
+		\end{pmatrix} = \det\begin{pmatrix}
+			u \\ w
+		\end{pmatrix} + k\det\begin{pmatrix}
+			v \\ w
+		\end{pmatrix}\]
+
 	and
-	
+
 	\[\det\begin{pmatrix}
-		w \\ u + kv
-	\end{pmatrix} = \det\begin{pmatrix}
-	w \\ u
-	\end{pmatrix} + k \det \begin{pmatrix}
-	w \\ v
-	\end{pmatrix}.\]
+			w \\ u + kv
+		\end{pmatrix} = \det\begin{pmatrix}
+			w \\ u
+		\end{pmatrix} + k \det \begin{pmatrix}
+			w \\ v
+		\end{pmatrix}.\]
 \end{theorem}
 
 \begin{theorem}\label{Theorem 4.2}
 	\hfill\\
 	Let $A \in M_{2 \times 2}(\F)$. Then the determinant of $A$ is nonzero if and only if $A$ is invertible. Moreover, if $A$ is invertible, then
-	
+
 	\[A^{-1} = \frac{1}{\det(A)}\begin{pmatrix}
-		A_{22} & -A_{12} \\
-		-A_{21} & A_{11}
-	\end{pmatrix}.\]
+			A_{22}  & -A_{12} \\
+			-A_{21} & A_{11}
+		\end{pmatrix}.\]
 \end{theorem}
 
 \begin{definition}
@@ -53,14 +53,14 @@
 \begin{definition}
 	\hfill\\
 	If $\beta = \{u,v\}$ is an ordered basis for $\R^2$, we define the \textbf{orientation} of $\beta$ to be the real number
-	
+
 	\[O\begin{pmatrix}
-		u \\ v
-	\end{pmatrix} = \frac{\det\begin{pmatrix}
-		u \\ v
-		\end{pmatrix}}{\abs{\det\begin{pmatrix}
 			u \\ v
-	\end{pmatrix}}}\]
+		\end{pmatrix} = \frac{\det\begin{pmatrix}
+				u \\ v
+			\end{pmatrix}}{\abs{\det\begin{pmatrix}
+					u \\ v
+				\end{pmatrix}}}\]
 
 	(The denominator of this fraction is nonzero by \autoref{Theorem 4.2}).
 \end{definition}
@@ -73,4 +73,4 @@
 \begin{definition}
 	\hfill\\
 	Any ordered set $\{u, v\}$ in $\R^2$ determines a parallelogram in the following manner. Regarding $u$ and $v$ as arrows emanating from the origin of $\R^2$, we call the parallelogram having $u$ and $v$ as adjacent sides the \textbf{parallelogram determined by $u$ and $v$}.
-\end{definition}
+\end{definition}