Finished all chapters and definitions. I need to add subsections and see if there's any theorems or definitions in the appendicies that are worth adding to this as well.

chapter-2/linear-transformations-null-spaces-and-ranges.tex

@@ -3,30 +3,30 @@
 \begin{definition}
 	\hfill\\
 	Let $V$ and $W$ be vector spaces (over $\F$). We call a function $T: V \to W$ a \textbf{linear transformation from $V$ to $W$} if, for all $x,y \in V$, and $c \in \F$, we have
-	
+
 	\begin{enumerate}
 		\item $T(x + y) = T(x) + T(y)$, and
 		\item $T(cx) = cT(x)$
 	\end{enumerate}
-	
+
 	If the underlying field $\F$ is the field of rational numbers, then (1) implies (2), but, in general (1) and (2) are logically independent.\\
-	
+
 	We often simply call $T$ \textbf{linear}.
 \end{definition}
 
 \begin{remark}
 	\hfill\\
 	Let $V$ and $W$ be vector spaces (over $\F$). Let $T: V \to W$ be a linear transformation. Then the following properties hold:
-	
+
 	\begin{enumerate}
 		\item If $T$ is linear, then $T(0) = 0$.
 		\item $T$ is linear if and only if $T(cx + y) = cT(x) + T(y)$ for all $x,y \in V$ and $c \in \F$.
 		\item If $T$ is linear, then $T(x-y)=T(x)-T(y)$ for all $x,y \in V$.
 		\item $T$ is linear if and only if, for $x_1, x_2, \dots, x_n \in V$ and $a_1, a_2, \dots, a_n \in \F$, we have
-		
-		\[T\left(\sum_{i=1}^{n}a_ix_i\right)=\sum_{i=1}^{n}a_iT(x_i).\]
+
+		      \[T\left(\sum_{i=1}^{n}a_ix_i\right)=\sum_{i=1}^{n}a_iT(x_i).\]
 	\end{enumerate}
-	
+
 	We generally use property 2 to prove that a given transformation is linear.
 \end{remark}
 
@@ -43,17 +43,17 @@
 \begin{definition}
 	\hfill\\
 	For vector spaces $V$ and $W$ (over $\F$), we define the \textbf{identity transformation} $I_V: V \to V$ by $I_V(x) = x$ for all $x \in V$.\\
-	
+
 	We define the \textbf{zero transformation} $T_0: V \to W$ by $T_0(x) = 0$ for all $x \in V$.\\
-	
+
 	\textbf{Note:} We often write $I$ instead of $I_V$.
 \end{definition}
 
 \begin{definition}
 	\hfill\\
-	Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. We define the \textbf{null space} (or \textbf{kernel}) $\n{T}$ to be the set of all vectors $x \in V$ such that $T(x)=0$; that is, \\$\n{T} = \{x \in V\ |\ T(x) = 0\}$.
-	
-	We define the \textbf{range} (or \textbf{image}) $\range{T}$ of $T$ to be the subset of $W$ consisting of all images (under $T$) of vectors in $V$; that is, $\range{T} = \{T(x)\ |\ x \in V\}$.
+	Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. We define the \textbf{null space} (or \textbf{kernel}) $\n{T}$ to be the set of all vectors $x \in V$ such that $T(x)=0$; that is, \\$\n{T} = \{x \in V : T(x) = 0\}$.
+
+		We define the \textbf{range} (or \textbf{image}) $\range{T}$ of $T$ to be the subset of $W$ consisting of all images (under $T$) of vectors in $V$; that is, $\range{T} = \{T(x) : x \in V\}$.
 \end{definition}
 
 \begin{theorem}
@@ -64,7 +64,7 @@
 \begin{theorem}
 	\hfill\\
 	Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. If $\beta = \{v_1, v_2, \dots, v_n\}$ is a basis for $V$, then
-	
+
 	\[\range{T} = \lspan{T(\beta)} = \lspan{\{T(v_1), T(v_2), \dots, T(v_n)\}}.\]
 \end{theorem}
 
@@ -76,7 +76,7 @@
 \begin{theorem}[\textbf{Dimension Theorem}]
 	\hfill\\
 	Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. If $V$ is finite-dimensional, then
-	
+
 	\[\nullity{T} + \rank{T} = \ldim{V}\]
 \end{theorem}
 
@@ -88,7 +88,7 @@
 \begin{theorem}\label{Theorem 2.5}
 	\hfill\\
 	Let $V$ and $W$ be vector spaces of equal (finite) dimension, and let $T: V \to W$ be linear. Then the following are equivalent.
-	
+
 	\begin{enumerate}
 		\item $T$ is one-to-one.
 		\item $T$ is onto.
@@ -114,4 +114,4 @@
 \begin{definition}
 	\hfill\\
 	Let $V$ be a vector space, and let $T: V \to W$ be linear. A subspace $W$ of $V$ is said to be \textbf{$T$-invariant} if $T(x) \in W$ for every $x \in W$, that is, $T(W) \subseteq W$. If $W$ is $T$-invariant, we define the \textbf{restriction of $T$ on $W$} to be the function $T_W: W \to W$ defined by $T_W(x) = T(x)$ for all $x \in W$.
-\end{definition}
+\end{definition}