Created the Abstract Algebra theorems and definitions cheat sheet

part-4/chapters/chapter-19/chapter-19.tex

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+\chapter{Vector Spaces}
+\subimport{./}{definition-and-examples.tex}
+\subimport{./}{subspaces.tex}
+\subimport{./}{linear-independence.tex}

part-4/chapters/chapter-19/definition-and-examples.tex

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+\section{Definition and Examples}
+
+\begin{definition}[Vector Space]
+	A set $V$ is said to be a \textit{vector space} over a field $\F$ if $V$ is an Abelian group under addition (denoted by $+$) and, if for each $a \in \F$ and $v \in V$, there is an element $av \in V$ such that the following conditions hold for all $a,b \in \F$ and all $u,v \in V$.
+	\begin{enumerate}
+		\item $a(v + u) = av + au$
+		\item $(a + b)v = av + bv$
+		\item $a(bv)=(ab)v$
+		\item $1v=v$
+	\end{enumerate}
+\end{definition}
+
+\begin{remark}
+	The members of a vector space are called \textit{vectors}. The members of the field are called \textit{scalars}. The operation that combines a scalar $a$ and a vector $v$ to form the vector $av$ is called \textit{scalar multiplication}. In general, we will denote vectors by letters from the end of the alphabet, such as $u,v,w$, and scalars by letters from the beginning of the alphabet, such as $a,b,c$.
+\end{remark}

part-4/chapters/chapter-19/linear-independence.tex

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+\section{Linear Independence}
+
+\begin{definition}[Linearly Dependent, Linearly Independent]
+	A set $S$ of vectors is said to be \textit{linearly dependent} over a field $\F$ if there are vectors $v_1,v_2,\dots,v_n$ from $S$ and elements $a_1,a_2,\dots,a_n$ from $\F$, not all zero, such that $a_1v_1+a_2v_2+\dots+a_nv_n = 0$. A set of vectors that is not linearly dependent over $\F$ is called \textit{linearly independent} over $\F$.
+\end{definition}
+
+\begin{definition}[Basis]
+	Let $V$ be a vector space over $\F$. A subset $B$ of $V$ is called a \textit{basis} for $V$ if $B$ is linearly independent over $\F$ and every element of $V$ is a linear combination of elements of $B$.
+\end{definition}
+
+\begin{theorem}[Invariance of Basis Size]
+	If $\{u_1,u_2,\dots,u_m\}$ and $\{w_1,w_2,\dots,w_n\}$ are both bases of a vector space $V$ over a field $\F$, then $m=n$.
+\end{theorem}
+
+\begin{definition}[Dimension]
+	A vector space that has a basis consisting of $n$ elements is said to have \textit{dimension $n$}. For completeness, the trivial vector space $\{0\}$ is said to be spanned by the empty set and to have dimension 0.
+
+	\noindent A vector space that has a finite basis is called \textit{finite dimensional}; otherwise, it is called \textit{infinite dimensional}.
+\end{definition}

part-4/chapters/chapter-19/subspaces.tex

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+\section{Subspaces}
+
+\begin{definition}[Subspace]
+	Let $V$ be a vector space over a field $\F$ and let $U$ be a subset of $V$. We say that $U$ is a \textit{subspace} of $V$ if $U$ is also a vector space over $\F$ under the operations of $V$.
+\end{definition}