Added subsections when they appear, added all of the appendices, and finished the packet

chapter-7/the-rational-canonical-form.tex

@@ -1,4 +1,4 @@
-\section{The rational Canonical Form}
+\section{The Rational Canonical Form}
 
 \begin{definition}
 	\hfill\\
@@ -174,6 +174,9 @@
 	Notice that $\alpha_j$ contains $p_jd$ vectors.
 \end{definition}
 
+\subsection*{Uniqueness of the Rational Canonical Form}
+\addcontentsline{toc}{subsection}{Uniqueness of the Rational Canonical Form}
+
 \begin{lemma}
 	\hfill\\
 	$\alpha_j$ is an ordered basis for $\mathsf{C}_{v_j}$.
@@ -211,6 +214,9 @@
 	Let $A \in M_{n \times n}(\F)$. The \textbf{rational canonical form} of $A$ is defined to be the rational canonical form of $L_A$. Likewise, for $A$, the \textbf{elementary divisors} and their \textbf{multiplicities} are the same as those of $L_A$.
 \end{definition}
 
+\subsection*{Direct Sums}
+\addcontentsline{toc}{subsection}{Direct Sums}
+
 \begin{theorem}[\textbf{Primary Decomposition Theorem}]
 	\hfill\\
 	Let $T$ be a linear operator on an $n$-dimensional vector space $V$ with characteristic polynomial