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/properties-of-determinants.tex

@@ -3,7 +3,7 @@
 \begin{remark}
 	\hfill\\
 	Because the determinant of the $n \times n$ matrix is $1$, we can interpret \autoref{Remark 4.1} as the following facts about the determinants of elementary matrices.
-	
+
 	\begin{enumerate}
 		\item If $E$ is an elementary matrix obtained by interchanging any two rows of $I$, then $\det(E) = -1$.
 		\item If $E$ is an elementary matrix obtained by multiplying some row of $I$ by the nonzero scalar $k$, then $\det(E) = k$.
@@ -28,10 +28,10 @@
 
 \begin{theorem}[\textbf{Cramer's Rule}]
 	\hfill\\
-	Let $Ax = b$ be the matrix form of a system of $n$ linear equations in $n$ unknowns, where $x = (x_1, x_2, \dots, x_n)^t$. If $\det(A) \neq 0$, then this system has a unique solution, and for each $k$ ($k = 1, 2, \dots, n$), 
-	
+	Let $Ax = b$ be the matrix form of a system of $n$ linear equations in $n$ unknowns, where $x = (x_1, x_2, \dots, x_n)^t$. If $\det(A) \neq 0$, then this system has a unique solution, and for each $k$ ($k = 1, 2, \dots, n$),
+
 	\[x_k = \frac{\det(M_k)}{\det(A)},\]
-	
+
 	where $M_k$ is the $n \times n$ matrix obtained from $A$ by replacing column $k$ of $A$ by $b$.
 \end{theorem}
 
@@ -68,14 +68,14 @@
 \begin{definition}
 	\hfill\\
 	A matrix of the form
-	
+
 	\[\begin{pmatrix}
-		1 & c_0 & c_0^2 & \dots & c_0^n \\
-		1 & c_1 & c_1^2 & \dots & c_1^n \\
-		\vdots & \vdots & \vdots & &\vdots \\
-		1 & c_n & c_n^2 & \dots & c_n^n
-	\end{pmatrix}\]
-	
+			1      & c_0    & c_0^2  & \dots & c_0^n  \\
+			1      & c_1    & c_1^2  & \dots & c_1^n  \\
+			\vdots & \vdots & \vdots &       & \vdots \\
+			1      & c_n    & c_n^2  & \dots & c_n^n
+		\end{pmatrix}\]
+
 	is called a \textbf{Vandermonde matrix}.
 \end{definition}
 
@@ -92,13 +92,13 @@
 \begin{definition}
 	\hfill\\
 	Let $y_1, y_2, \dots, y_n$ be linearly independent function in $\C^\infty$. For each $y \in \C^\infty$, define $T(y) \in \C^\infty$ by
-	
+
 	\[[T(y)](t) = \det\begin{pmatrix}
-		y(t) & y_1(t) & y_2(t) & \dots & y_n(t) \\
-		y'(t) & y'_1(t) & y'_2(t) & \dots & y'_n(t) \\
-		\vdots & \vdots & \vdots & &\vdots \\
-		y^{(n)}(t) & y_1^{(n)}(t) & y_2^{(n)}(t) & \dots & y_n^{(n)}(t)
-	\end{pmatrix}\]
-	
+			y(t)       & y_1(t)       & y_2(t)       & \dots & y_n(t)       \\
+			y'(t)      & y'_1(t)      & y'_2(t)      & \dots & y'_n(t)      \\
+			\vdots     & \vdots       & \vdots       &       & \vdots       \\
+			y^{(n)}(t) & y_1^{(n)}(t) & y_2^{(n)}(t) & \dots & y_n^{(n)}(t)
+		\end{pmatrix}\]
+
 	The preceding determinant is called the \textbf{Wronskian} of $y, y_1, \dots, y_n$.
-\end{definition}
+\end{definition}