Wrote out chapters 2-4

chapter-1/linear-combinations-and-systems-of-linear-equations.tex

@@ -7,7 +7,7 @@
 
 \begin{definition}
 	\hfill\\
-	Let $S$ be a nonempty subset of a vector space $V$. The \textbf{span} of $S$, denoted $\text{span}(S)$, is the set consisting of all linear combinations of the vectors in $S$. For convenience, we define $\text{span}(\emptyset) = \{0\}$.
+	Let $S$ be a nonempty subset of a vector space $V$. The \textbf{span} of $S$, denoted $\lspan{S}$, is the set consisting of all linear combinations of the vectors in $S$. For convenience, we define $\lspan{\emptyset} = \{0\}$.
 \end{definition}
 
 \begin{theorem}
@@ -18,4 +18,4 @@
 \begin{definition}
 	\hfill\\
 	A subset $S$ of a vector space $V$ \textbf{generates} (or \textbf{spans}) $V$ if $\text{span}(S) = V$. In this case, we also say that the vectors of $S$ generate (or span) $V$.
-\end{definition}
+\end{definition}

chapter-1/linear-dependence-and-linear-independence.tex

@@ -3,9 +3,9 @@
 \begin{definition}
 	\hfill\\
 	A subset $S$ of a vector space $V$ is called \textbf{linearly dependent} if there exist a finite number of distinct vectors $v_1, v_2, \dots, v_n$ in $S$ and scalars $a_1, a_2, \dots, a_n$ not all zero, such that
-	
+
 	\[a_1v_2 + a_2v_2 + \dots + a_nv_n = 0\]
-	
+
 	In this case, we also say that the vectors of $S$ are linearly dependent.\\
 
 	For any vectors $v_1, v_2, \dots, v_n$, we have $a_1v_1 + a_2v_2 + \dots + a_nv_n = 0$ if $a_1 = a_2 = \dots = a_n = 0$. We call this the \textbf{trivial representation} of $0$ as a linear combination of $v_1, v_2, \dots, v_n$. Thus, for a set tot be linearly dependent, there must exist a nontrivial representation of $0$ as a linear combination of vectors in the set. Consequently, any subset of a vector space that contains the zero vector is linearly dependent, because $0 = 1 \cdot 0$ is a nontrivial representation of $0$ as a linear combination of vectors in the set.
@@ -14,15 +14,15 @@
 \begin{definition}
 	\hfill\\
 	A subset $S$ of a vector space that is not linearly dependent is called \textbf{linearly independent}. As before, we also say that the vectors of $S$ are linearly independent.\\
-	
+
 	The following facts about linearly independent sets are true in any vector space.
-	
+
 	\begin{enumerate}
 		\item The empty set is linearly independent, for linearly dependent sets must be nonempty.
 		\item A set consisting of a single nonzero vector is linearly independent. For if $\{v\}$ is linearly dependent, then $av = 0$ for some nonzero scalar $a$. thus
-		
-		\[v = a^{-1}(av) = a^{-1}0 = 0.\]
-		
+
+		      \[v = a^{-1}(av) = a^{-1}0 = 0.\]
+
 		\item A set is linearly independent if and only if the only representations of $0$ as linear combinations of its vectors are trivial representations.
 	\end{enumerate}
 \end{definition}
@@ -39,5 +39,5 @@
 
 \begin{theorem}
 	\hfill\\
-	Let $S$ be a linearly independent subset of a vector space $V$, and let $v$ be a vector in $V$ that is not in $S$. Then $S \cup \{v\}$ is linearly dependent if and only if $v \in \text{span}(S)$.
-\end{theorem}
+	Let $S$ be a linearly independent subset of a vector space $V$, and let $v$ be a vector in $V$ that is not in $S$. Then $S \cup \{v\}$ is linearly dependent if and only if $v \in \lspan{S}$.
+\end{theorem}

chapter-1/vector-spaces.tex

@@ -60,7 +60,7 @@
 	for $1 \leq i \leq m$ and $1 \leq j \leq n$.
 \end{definition}
 
-\begin{definition}
+\begin{definition}\label{Definition 1.7}
 	\hfill\\
 	Let $S$ be any nonempty set and $\F$ be any field, and let $\mathcal{F}(S, \F)$ denote the set of all functions from $S$ to $\F$. Two functions $f$ and $g$ in $\mathcal{F}(S, \F)$ are called \textbf{equal} if $f(s) = g(s)$ for each $s \in S$. The set $\mathcal{F}(S, \F)$ is a vector space with the operations of addition and scalar multiplication defined for $f,g \in \mathcal{F}(S, \F)$ and $c \in \F$ defined by