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

chapter-6/bilinear-and-quadratic-forms.tex

@@ -1,5 +1,8 @@
 \section{Bilinear and Quadratic Forms}
 
+\subsection*{Bilinear Forms}
+\addcontentsline{toc}{subsection}{Bilinear Forms}
+
 \begin{definition}
 	\hfill\\
 	Let $V$ be a vector space over a field $\F$. A function $H$ from the set $V \times V$ if ordered pairs of vectors to $\F$ is called a \textbf{bilinear form} on $V$ if $H$ is linear in each variable when the other variable is held fixed; that is, $H$ is a bilinear form on $V$ if
@@ -76,6 +79,9 @@
 	Let $V$ be an $n$-dimensional vector space with ordered basis $\beta$, and let $H$ be a bilinear form on $V$. For any $n \times n$ matrix $B$, if $B$ is congruent to $\psi_\beta(H)$, then there exists an ordered basis $\gamma$ for $V$ such that $\psi_\gamma(H) = B$. Furthermore, if $B = Q^t\psi_\beta(H)Q$ for some invertible matrix $Q$, then $Q$ changes $\gamma$-coordinates into $\beta$-coordinates.
 \end{corollary}
 
+\subsection*{Symmetric Bilinear Forms}
+\addcontentsline{toc}{subsection}{Symmetric Bilinear Forms}
+
 \begin{definition}
 	\hfill\\
 	A bilinear form $H$ on a vector space $V$ is \textbf{symmetric} if $H(x,y) = H(y,x)$ for all $x,y \in V$.
@@ -116,6 +122,9 @@
 	Let $\F$ be a field that is not of characteristic two. If $A \in M_{n \times n}(\F)$ is a symmetric matrix, then $A$ is congruent to a diagonal matrix.
 \end{corollary}
 
+\subsection*{Quadratic Forms}
+\addcontentsline{toc}{subsection}{Quadratic Forms}
+
 \begin{definition}
 	\hfill\\
 	Let $V$ be a vector space over $\F$. A function $K: V \to \F$ is called a \textbf{quadratic form} if there exists a symmetric bilinear form $H \in \mathcal{B}(V)$ such that
@@ -123,6 +132,9 @@
 	\[K(x) = H(x, x)\ \ \ \text{for all}\ x \in V.\]
 \end{definition}
 
+\subsection*{Quadratic Forms Over the Field $\R$}
+\addcontentsline{toc}{subsection}{Quadratic Forms Over the Field $\R$}
+
 \begin{theorem}
 	\hfill\\
 	Let $V$ be a finite-dimensional real inner product space, and let $H$ be a symmetric bilinear form on $V$. Then there exists an orthonormal basis $\beta$ for $V$ such that $\psi_\beta(H)$ is a diagonal matrix.
@@ -141,6 +153,9 @@
 	In fact, if $H$ is the symmetric bilinear form determined by $K$, then $\beta$ can be chosen to be any orthonormal basis for $V$ such that $\psi_\beta(H)$ is a diagonal matrix.
 \end{corollary}
 
+\subsection*{The Second Derivative Test for Functions of Several Variables}
+\addcontentsline{toc}{subsection}{The Second Derivative Test for Functions of Several Variables}
+
 \begin{definition}
 	\hfill\\
 	Let $z=f(t_1, t_2, \dots, t_n)$ be a fixed real-valued function of $n$ real variables for which all third-order partial derivatives exist and are continuous. The function $f$ is said to have a \textbf{local maximum} at point $p \in \R^n$ if there exists a $\delta > 0$ such that $f(p) \geq f(x)$ whenever $||x - p|| < \delta$. Likewise, $f$ has a \textbf{local minimum} at $p \in \R^n$ if there exists a $\delta > 0$ such that $f(p) \leq f(x)$ whenever $||x - p|| < \delta$. If $f$ has either a local minimum or a local maximum at $p$, we say that $f$ has a \textbf{local extremum} at $p$. A point $p \in \R^n$ is called a \textbf{critical point} of $f$ if $\displaystyle\frac{\partial f(p)}{\partial(t_i)} = 0$ for $i = 1, 2, \dots, n$. It is a well known fact that if $f$ has a local extremum at a point $p \in \R^n$, then $p$ is a critical point of $f$. For, if $f$ has a local extremum at $p=(p_1, p_2, \dots, p_n)$, then for any $i = 1, 2, \dots, n$, the function $\phi_i$ defined by $\phi_i(t) = f(p_1, p_2, \dots, p_{i-1}, t, p_{i+1}, \dots, p_n)$ has a local extremum at $t = p_i$. So, by an elementary single-variable argument,
@@ -168,6 +183,9 @@
 	\end{enumerate}
 \end{theorem}
 
+\subsection*{Sylvester's Law of Inertia}
+\addcontentsline{toc}{subsection}{Sylvester's Law of Inertia}
+
 \begin{definition}
 	\hfill\\
 	The \textbf{rank} of a bilinear form is the rank of any of its matrix representations.