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