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

chapter-5/diagonalizability.tex

@@ -63,7 +63,10 @@
 	\end{enumerate}
 \end{theorem}
 
-\begin{remark}[\textbf{Test for Diagonalization}]
+\subsection*{Test for Diagonalization}
+\addcontentsline{toc}{subsection}{Test for Diagonalization}
+
+\begin{remark}
 	\hfill\\
 	Let $T$ be a linear operator on an $n$-dimensional vector space $V$. Then $T$ is diagonalizable if and only if both of the following conditions hold.
 

chapter-5/invariant-subspaces-and-the-cayley-hamilton-theorem.tex

@@ -29,6 +29,9 @@
 	\end{enumerate}
 \end{theorem}
 
+\subsection*{The Cayley-Hamilton Theorem}
+\addcontentsline{toc}{subsection}{The Cayley-Hamilton Theorem}
+
 \begin{theorem}[\textbf{Cayley-Hamilton}]
 	\hfill\\
 	Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $f(t)$ be the characteristic polynomial of $T$. Then $f(T) = T_0$, the zero transformation. That is, $T$ ``satisfies" its characteristic equation.
@@ -39,6 +42,9 @@
 	Let $A$ be an $n \times n$ matrix, and let $f(t)$ be the characteristic polynomial of $A$. Then $f(A) = O$, the $n \times n$ zero matrix.
 \end{corollary}
 
+\subsection*{Invariant Subspaces and Direct Sums}
+\addcontentsline{toc}{subsection}{Invariant Subspaces and Direct Sums}
+
 \begin{theorem}
 	\hfill\\
 	Let $T$ be a linear operator on a finite-dimensional vector space $V$, and suppose that $V = W_1 \oplus W_2 \oplus \dots \oplus W_k$, where $W_i$ is a $T$-invariant subspace of $V$ for each $i$ ($1 \leq i \leq k$). Suppose that $f_i(t)$ is the characteristic polynomial of $T_{W_i}$ ($1 \leq i \leq k$). Then $f_1(t)\cdot f_2(t) \cdot \dots \cdot f_k(t)$ is the characteristic polynomial of $T$.