Linear Algebra edits

Advanced/Linear Algebra 101/parts/1 vectors.tex

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+\section{Vectors}
+
+\definition{}
+Elements of $\mathbb{R}^n$ are often called \textit{vectors}. \\
+As you may already know, we have a few operations on vectors:
+\begin{itemize}
+	\item Vector addition: $[a_1, a_2] + [b_1, b_2] = [a_1+b_1, a_2+b_2]$
+	\item Scalar multiplication: $x \times [a_1, a_2] = [xa_1, xa_2]$.
+\end{itemize}
+\note{
+	The above examples are for $\mathbb{R}^2$, and each vector thus has two components. \\
+	These operations are similar for all other $n$.
+}
+
+\problem{}
+Compute the following or explain why you can't:
+\begin{itemize}
+	\item $[1, 2, 3] - [1, 3, 4]$ \note{Subtraction works just like addition.}
+	\item $4 \times [5, 2, 4]$
+	\item $a + b$, where $a \in \mathbb{R}
+^5$ and $b \in \mathbb{R}^7$
+\end{itemize}
+
+\vfill
+
+\problem{}
+Consider $(2, -1)$ and $(3, 1)$ in $\mathbb{R}^2$. \\
+Can you develop geometric intuition for their sum and difference?
+
+\begin{center}
+	\begin{tikzpicture}[scale=1]
+
+		\draw[->]
+			(0,0) coordinate (o) -- node[below left] {$(1, 2)$}
+			(2, -1) coordinate (a)
+		;
+
+		\draw[->]
+			(a) -- node[below right] {$(3, 1)$}
+			(5, 0) coordinate (b)
+		;
+
+		\draw[
+			draw = gray,
+			text = gray,
+			->
+		]
+			(o) -- node[above] {$??$}
+			(b) coordinate (s)
+		;
+
+	\end{tikzpicture}
+	\end{center}
+
+
+\vfill
+\pagebreak
+
+\definition{Euclidean Norm}
+In general, a \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
+Usually, one thinks of a norm as a \say{length metric} on a vector space. \\
+The norm of a vector $v$ is written $||v||$. \\
+
+\vspace{2mm}
+
+We usually use the \textit{euclidean norm} when we work in $\mathbb{R}^n$. \\
+If $v \in \mathbb{R}^n$, the euclidean norm is defined as follows:
+$$
+	||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}
+$$
+This is simply an application of the pythagorean theorem.
+
+\problem{}
+Compute the euclidean norm of
+\begin{itemize}
+	\item $[2, 3]$
+	\item $[-2, 1, -4, 2]$
+\end{itemize}
+
+\vfill
+\pagebreak