Linear Algebra edits

Advanced/Linear Algebra 101/parts/0 notation.tex

@@ -0,0 +1,87 @@
+\section{Notation and Terminology}
+
+\definition{}
+\begin{itemize}
+	\item $\mathbb{R}$ is the set of all real numbers.
+	\item $\mathbb{R}^+$ is the set of positive real numbers. Zero is not positive.
+	\item $\mathbb{R}^+_0$ is the set of positive real numbers and zero.
+\end{itemize}
+
+Mathematicians are often inconsistent with their notation. Depending on the author, their mood, and the phase of the moon, $\mathbb{R}^+$ may or may not include zero. I will use the definitions above.
+
+
+\definition{}
+Consider two sets $A$ and $B$. The set $A \times B$ consists of all tuples $(a, b)$ where $a \in A$ and $b \in B$. \\
+For example, $\{1, 2, 3\} \times \{\heartsuit, \star\} = \{(1,\heartsuit), (1, \star), (2,\heartsuit), (2, \star), (3,\heartsuit), (3, \star)\}$ \\
+This is called the \textit{cartesian product}.
+
+\vspace{4mm}
+
+You can think of this as placing the two sets \say{perpendicular} to one another:
+
+\begin{center}
+\begin{tikzpicture}[
+	scale=1,
+	bullet/.style={circle,inner sep=1.5pt,fill}
+]
+	\draw[->] (-0.2,0) -- (4,0) node[right]{$A$};
+	\draw[->] (0,-0.2) -- (0,3) node[above]{$B$};
+
+	\draw (1,0.1) -- ++ (0,-0.2) node[below]{$1$};
+	\draw (2,0.1) -- ++ (0,-0.2) node[below]{$2$};
+	\draw (3,0.1) -- ++ (0,-0.2) node[below]{$3$};
+
+	\draw (0.1, 1) -- ++ (-0.2, 0) node[left]{$\heartsuit$};
+	\draw (0.1, 2) -- ++ (-0.2, 0) node[left]{$\star$};
+
+	\node[bullet] at (1, 1){};
+	\node[bullet] at (2, 1) {};
+	\node[bullet] at (3, 1) {};
+	\node[bullet] at (1, 2) {};
+	\node[bullet] at (2, 2) {};
+	\node[bullet] at (3, 2) {};
+
+
+	\draw[rounded corners] (0.5, 0.5) rectangle (3.5, 2.5) {};
+	\node[above] at (2, 2.5) {$A \times B$};
+
+\end{tikzpicture}
+\end{center}
+
+\problem{}
+Let $A = \{0, 1\} \times \{0, 1\}$ \\
+Let $B = \{ a, b\}$ \\
+What is $A \times B$?
+
+\vfill
+
+\problem{}
+What is $\mathbb{R} \times \mathbb{R}$? \\
+\hint{Use the \say{perpendicular} analogy}
+
+\vfill
+\pagebreak
+
+\definition{}
+$\mathbb{R}^n$ is the set of $n$-tuples of real numbers. \\
+In english, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \\
+
+\vspace{4mm}
+
+Elements of $\mathbb{R}^2$ look like $(a, b)$, where $a, b \in \mathbb{R}$. \hfill \note{\textit{Note:} $\mathbb{R}^2$ is pronounced \say{arrgh-two.}}
+Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4 a_5)$, where $a_n \in \mathbb{R}$. \\
+
+$\mathbb{R}^1$ and $\mathbb{R}$ are identical.
+
+\vspace{4mm}
+
+Intuitively, $\mathbb{R}^2$ forms a two-dimensional plane, and $\mathbb{R}^3$ forms a three-dimensional space. \\
+$\mathbb{R}^n$ is hard to visualize when $n \geq 4$, but you are welcome to try.
+
+\problem{}
+Convince yourself that $\mathbb{R} \times \mathbb{R}$ is $\mathbb{R}^2$. \\
+What is $\mathbb{R}^2 \times \mathbb{R}$?
+
+
+\vfill
+\pagebreak

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

@@ -0,0 +1,81 @@
+\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