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

chapter-6/the-adjoint-of-a-linear-operator.tex

@@ -58,6 +58,9 @@
 	\end{enumerate}
 \end{corollary}
 
+\subsection*{Least Squares Approximation}
+\addcontentsline{toc}{subsection}{Least Squares Approximation}
+
 \begin{lemma}
 	\hfill\\
 	Let $A \in M_{m \times n}(\F), x \in F^n$, and $y \in F^m$. Then
@@ -85,6 +88,9 @@
 	A solution $s$ to a system of linear equations $Ax = b$ is called a \textbf{minimal solution} if $||s|| \leq ||u||$ for all other solutions $u$.
 \end{definition}
 
+\subsection*{Minimal Solutions to Systems of Linear Equations}
+\addcontentsline{toc}{subsection}{Minimal Solutions to Systems of Linear Equations}
+
 \begin{theorem}
 	\hfill\\
 	Let $A \in M_{m \times n}(\F)$ and $b \in F^m$. Suppose that $Ax = b$ is consistent. Then the following statements are true.