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

\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. We 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