Added linear maps handout

Advanced/Linear Maps/parts/1 spaces.tex

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+\definition{Vector Spaces}
+A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
+\begin{itemize}[itemsep = 2mm]
+	\item A set $V$, the elements of which are called \textit{vectors}
+	\item An operation called \textit{vector addition}, denoted $+$ \\
+		Vector addition operates on two elements of $V$. \\
+
+	\item An operation called \textit{scalar multilplication}, denoted $\times$ \\
+		Scalar multiplication multiplies an element of $V$ by an element of $\mathbb{F}$. \\
+		Any element of $\mathbb{F}$ is called a \textit{scalar}.
+\end{itemize}
+
+\vspace{2mm}
+
+\textbf{Note:}
+The same symbols are used for additions and multiplications in both $\mathbb{F}$ and $V$. \\
+Be careful, since \textit{these are different operations!} \\
+Make sure you're aware of the context of each $+$ and $\times$ as you work through this handout.
+
+\vspace{5mm}
+
+Vector addition and multiplication must have the following properties. \\
+Note that $x, y, z \in V$ and $a, b\in \mathbb{F}$.
+
+\vspace{2mm}
+
+% [t] and \vspace{0pt} ensure alignment at top
+\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
+	\begin{center}
+	\begin{tabular}{l | r@{=}l }
+		\hline
+		\multicolumn{3}{|c|}{Properties of vector addition} \\
+		\hline
+		Closure			& \multicolumn{2}{c}{$x+y \in V$} \\
+		Associativity	& $(x+y)+z~$&$~x+y+z$	\\
+		Commutativity	& $x+y~$&$~y+x$ 		\\
+		Distributivity	& $x(y+z)~$&$~xy + xz$	\\
+		Identity		& $x+0~$&$~x$			\\
+		Inverse			& $x + (-x)~$&$~0$
+	\end{tabular}
+	\end{center}
+\end{minipage}%
+\hfill%
+\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
+	\begin{center}
+	\begin{tabular}{l | r@{=}l }
+		\hline
+		\multicolumn{3}{|c|}{Properties of vector multiplication} \\
+		\hline
+		Closure			& \multicolumn{2}{c}{$ax \in V$} \\
+		Distributivity	& $a(x+y)~$&$~ax+ay$	\\
+						& $(a+b)x~$&$~ax+bx$	\\
+		Compatibility$^*$	& $(ab)x~$&$~x(ba)$		\\
+		Identity		& $a+0~$&$~a$
+	\end{tabular}
+\end{center}
+\end{minipage}
+
+\vspace{5mm}
+
+\definition{}
+There is a good chance you are familiar with basic vector arithmetic. \\
+Here's a quick review:
+\begin{itemize}
+	\item Scalar multiplication is done elementwise: $3 \times [a, b, c] = [3a, 3b, 3c]$.
+	\item Vector addition is similar: $[a, b, c] + [1, 2, 3] = [a+1,~b+2,~c+3]$.
+	\item Vector addition is not valid for vectors of different sizes.
+\end{itemize}
+
+
+\definition{}
+We usually use the \textit{dot product} as our vector product. It is defined as follows. \\
+Given two vectors $a, b \in \mathbb{R}^n$, the dot product $a \cdot b$ is $\sum_1^n a_ib_i$.
+
+\vspace{2mm}
+
+In other words, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$,
+$$
+	a \cdot b = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32
+$$
+As you may expect, the dot product $ab$ is valid iff $a$ and $b$ are the same size.
+
+
+
+\problem{}
+Show that the dot product satisfies the properties of a vector product listed above. \\
+Conclude that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$.
+
+\vfill
+\pagebreak
+
+