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-1/subspaces.tex

@@ -3,11 +3,11 @@
 \begin{definition}
 	\hfill\\
 	A subset $W$ of a vector space $V$ over a field $\F$ is called a \textbf{subspace} of $V$ if $W$ is a vector space over $\F$ with the operations of addition and scalar multiplication defined on $V$.\\
-	
+
 	In any vector space $V$, note that $V$ and $\{0\}$ are subspaces. The latter is called the \textbf{zero subspace} of $V$.
-	
+
 	Fortunately, it is not necessary to verify all of the vector space properties to prove that a subset is a subspace. Because properties (VS 1), (VS 2), (VS 5), (VS 6), (VS 7) and (VS 8) hold for all vectors in the vector space, these properties automatically hold for the vectors in any subset. Thus a subset $W$ of a vector space $V$ is a subspace of $V$ if and only if the following four properties hold:
-	
+
 	\begin{enumerate}
 		\item $x + y \in W$ whenever $x \in W$ and $y \in W$. ($W$ is \textbf{closed under addition}).
 		\item $cx \in W$ whenever $c \in \F$ and $x \in W$. ($W$ is \textbf{closed under scalar multiplication}).
@@ -19,7 +19,7 @@
 \begin{theorem}
 	\hfill\\
 	Let $V$ be a vector space and $W$ a subset of $V$. Then $W$ is a subspace of $V$ if and only if the following three conditions hold for the operations defined in $V$.
-	
+
 	\begin{enumerate}
 		\item $0 \in W$.
 		\item $x + y \in W$ whenever $x \in W$ and $y \in W$.
@@ -45,7 +45,7 @@
 \begin{definition}
 	\hfill\\
 	The \textbf{trace} of an $n \times n$ matrix $M$, denoted $\text{tr}(M)$, is the sum of the diagonal entries of $M$; that is,
-	
+
 	\[\text{tr}(M) = M_{11} + M_{22} + \dots + M_{nn}.\]
 \end{definition}
 
@@ -57,4 +57,4 @@
 \begin{definition}
 	\hfill\\
 	An $m \times n$ matrix $A$ is called \textbf{upper triangular} if all entries lying below the diagonal entries are zero; that is, if $A_{ij} = 0$ whenever $i > j$.
-\end{definition}
+\end{definition}