Edits & renames

Advanced/Introduction to Quantum/parts/00 vectors.tex

@@ -0,0 +1,110 @@
+\section*{Prerequisite: Vector Basics}
+
+\definition{Vectors}
+An $n$-dimensional \textit{vector} is an element of $\mathbb{R}^n$. In this handout, we'll write vectors as columns. \par
+For example, $\left[\begin{smallmatrix} 1 \\ 3 \\ 2 \end{smallmatrix}\right]$ is a vector in $\mathbb{R}^3$.
+
+
+\definition{Euclidean norm}
+The length of an $n$-dimensional vector $v$ is computed as follows:
+\begin{equation*}
+	|v| = \sqrt{v_0^2 +v_1^2 + ... + v_n^2}
+\end{equation*}
+Where $v_0$ through $v_n$ represent individual components of this vector. For example,
+\begin{equation*}
+	\left|\left[\begin{smallmatrix} 1 \\ 3 \\ 2 \end{smallmatrix}\right]\right| = \sqrt{1^2 + 3^2 + 2^2} = \sqrt{14}
+\end{equation*}
+
+\definition{Transpose}
+The \textit{transpose} of a vector $v$ is $v^\text{T}$, given as follows:
+\begin{equation*}
+	\left[\begin{smallmatrix} 1 \\ 3 \\ 2 \end{smallmatrix}\right]^\text{T}
+	=
+	\left[\begin{smallmatrix} 1 & 3 & 2 \end{smallmatrix}\right]
+\end{equation*}
+That is, we rewrite the vector with its rows as columns and its columns as rows. \par
+We can transpose matrices too, of course, but we'll get to that later.
+
+
+\problem{}
+What is the length of $\left[\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right]^\text{T}$? \par
+
+\vfill
+
+\definition{}
+We say a vector $v$ is a \textit{unit vector} or a \textit{normalized} vector if $|v| = 1$.
+\pagebreak
+
+
+\definition{Vector products}
+The \textit{dot product} of two $n$-dimensional vectors $v$ and $u$ is computed as follows:
+\begin{equation*}
+	v \cdot u = v_0u_0 + v_1u_1 + ... + v_nu_n
+\end{equation*}
+
+
+\vfill
+
+\definition{Vector angles}<vectorangle>
+For any two vectors $a$ and $b$, the following holds:
+
+\null\hfill
+\begin{minipage}{0.48\textwidth}
+	\begin{equation*}
+		\cos{(\phi)} = \frac{a \cdot b}{|a| \times |b|}
+	\end{equation*}
+\end{minipage}
+\hfill
+\begin{minipage}{0.48\textwidth}
+	\begin{center}
+		\begin{tikzpicture}[scale=1.5]
+			\draw[->] (0, 0) -- (0.707, 0.707);
+			\draw[->, gray] (0.5, 0.0) arc (0:45:0.5);
+			\node[gray] at (0.6, 0.22) {$\phi$};
+
+			\draw[->] (0, 0) -- (1.2, 0);
+			\node[right] at (1.2, 0) {$a$};
+
+			\node[right] at (0.707, 0.707) {$b$};
+		\end{tikzpicture}
+	\end{center}
+\end{minipage}
+\hfill\null
+This can easily be shown using the law of cosines. \par
+For the sake of time, we'll skip the proof---it isn't directly relevant to this handout.
+
+\definition{Orthogonal vectors}
+We say two vectors are \textit{perpendicular} or \textit{orthogonal} if the angle between them is $90^\circ$. \par
+Note that this definition works with vectors of any dimension.
+
+\note{
+	In fact, we don't need to think about other dimensions: two vectors in an $n$-dimensional space nearly always
+	define a unique two-dimensional plane (with two exceptions: $\phi = 0^\circ$ and $\phi = 180^\circ$).
+}
+
+
+\problem{}
+What is the dot product of two orthogonal vectors?
+
+
+\vfill
+\pagebreak
+
+
+%For example, the set $\{[1,0,0], [0,1,0], [0,0,1]\}$ (which we usually call $\{x, y, z\})$
+%forms an orthonormal basis of $\mathbb{R}^3$. Every element of $\mathbb{R}^3$ can be written as a linear combination of these vectors:
+%
+%\begin{equation*}
+%	\left[\begin{smallmatrix} a \\ b \\ c \end{smallmatrix}\right]
+%	=
+%	a \left[\begin{smallmatrix} 1 \\ 0 \\ 0 \end{smallmatrix}\right] +
+%	b \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \end{smallmatrix}\right] +
+%	c \left[\begin{smallmatrix} 0 \\ 0 \\ 1 \end{smallmatrix}\right]
+%\end{equation*}
+%
+%The tuple $[a,b,c]$ is called the \textit{coordinate} of a point with respect to this basis.
+
+
+
+\vfill
+\pagebreak

Advanced/Introduction to Quantum/parts/01 bits.tex

@@ -142,56 +142,13 @@ We could, of course, mark the point \texttt{x} at $[1, 1]$, which is equal parts
 \vspace{4mm}
 
 But \texttt{x} isn't a member of $\mathbb{B}$, it's not a valid state. \par
-Our bit is fully $\vec{e}_0$ or fully $\vec{e}_1$. By our current definitions, there's nothing in between.
+Our bit is fully $\vec{e}_0$ or fully $\vec{e}_1$. By our current definitions, there's nothing in between. \par
-
-
-\vspace{8mm}
-
-
-
-
-
-
-
-
-
-
-
-
-\definition{Orthonormal Basis}
-The unit vectors $\vec{e}_0$ and $\vec{e}_1$ form an \textit{orthonormal basis} of the plane $\mathbb{R}^2$. \par
 \note{
-	\say{ortho-} means \say{orthogonal}; normal means \say{normal,} which means length $= 1$. \\
+	Note that the unit vectors $\vec{e}_0$ and $\vec{e}_1$ form an \textit{orthonormal basis} of the plane $\mathbb{R}^2$.
-}{
+}
-	Note that $\vec{e}_0$ and $\vec{e}_1$ are orthonormal by \textit{definition}. \\
-	We don't have to prove anything, we simply defined them as such.
-} \par
-
-\vspace{2mm}
-
-There's much more to say about basis vectors, but we don't need all the tools of linear algebra here. \par
-We just need to understand that a set of $n$ orthogonal unit vectors defines an $n$-dimensional space. \par
-This is fairly easy to think about: each vector corresponds to an axis of the space, and every point
-in that space can be written as a \textit{linear combination} (i.e, a weighted sum) of these basis vectors.
-
-\vspace{2mm}
-
-For example, the set $\{[1,0,0], [0,1,0], [0,0,1]\}$ (which we usually call $\{x, y, z\})$
-forms an orthonormal basis of $\mathbb{R}^3$. Every element of $\mathbb{R}^3$ can be written as a linear combination of these vectors:
-
-\begin{equation*}
-	\left[\begin{smallmatrix} a \\ b \\ c \end{smallmatrix}\right]
-	=
-	a \left[\begin{smallmatrix} 1 \\ 0 \\ 0 \end{smallmatrix}\right] +
-	b \left[\begin{smallmatrix} 0 \\ 1 \\ 0 \end{smallmatrix}\right] +
-	c \left[\begin{smallmatrix} 0 \\ 0 \\ 1 \end{smallmatrix}\right]
-\end{equation*}
-
-The tuple $[a,b,c]$ is called the \textit{coordinate} of a point with respect to this basis.
-
-
 
 \vfill
+
 \pagebreak
 
 

Advanced/Introduction to Quantum/parts/02 two bits.tex

@@ -232,7 +232,8 @@ $?
 
 
 \problem{}
-The vectors we found in \ref{basistp} are a basis of what space? \par
+What is the \textit{span} of the vectors we found in \ref{basistp}? \par
+In other words, what is the set of vectors that can be written as weighted sums of the vectors above?
 
 \vfill
 \pagebreak