114 lines
3.6 KiB
TeX
114 lines
3.6 KiB
TeX
\section{Spaces}
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\definition{}
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A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
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\begin{itemize}[itemsep = 2mm]
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\item A set $V$, the elements of which are called \textit{vectors}
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\item An operation called \textit{vector addition}, denoted $+$ \\
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Vector addition operates on two elements of $V$. \\
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\item An operation called \textit{scalar multiplication}, denoted $\times$ \\
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Scalar multiplication multiplies an element of $V$ by an element of $\mathbb{F}$. \\
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Any element of $\mathbb{F}$ is called a \textit{scalar}.
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\end{itemize}
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\vspace{2mm}
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\textbf{Note:}
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The same symbols are used for additions and multiplications in both $\mathbb{F}$ and $V$. \\
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\textit{These are different operations}, so be aware of the context of each $+$ and $\times$.
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\vspace{5mm}
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Vector addition and multiplication must have the following properties. \\
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In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$.
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\vspace{2mm}
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% [t] and \vspace{0pt} ensure alignment at top
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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\begin{tabular}{l | r@{=}l }
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\hline
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\multicolumn{3}{|c|}{Properties of vector addition} \\
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\hline
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Closure & \multicolumn{2}{c}{$x+y \in V$} \\
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Associativity & $(x+y)+z~$&$~x+y+z$ \\
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Commutativity & $x+y~$&$~y+x$ \\
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Distributivity & $x(y+z)~$&$~xy + xz$ \\
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Identity & $x+0~$&$~x$ \\
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Inverse & $x + (-x)~$&$~0$
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\end{tabular}
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\end{center}
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\end{minipage}%
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\hfill%
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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\begin{tabular}{l | r@{=}l }
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\hline
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\multicolumn{3}{|c|}{Properties of scalar multiplication} \\
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\hline
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Closure & \multicolumn{2}{c}{$ax \in V$} \\
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Distributivity & $a(x+y)~$&$~ax+ay$ \\
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& $(a+b)x~$&$~ax+bx$ \\
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Compatibility$^*$ & $(ab)x~$&$~a(bx)$ \\
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Identity & $a+0~$&$~a$
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\end{tabular}
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\end{center}
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\end{minipage}
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\vspace{2mm}
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$^*$ Remember that $a, b \in \mathbb{F}$ and $x \in V$. Thus, $(ab)$ is multiplication in $\mathbb{F}$ and $(bx)$ is scalar multiplication in $V$. Compatibility is \textit{not} associativity. \\
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Likewise, the addition you see in the distributive property of multiplication is field addition, not vector addition.
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\vspace{6mm}
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Usually, the word \textit{vector} refers to an element of $\mathbb{R}^n$. As you might expect $\mathbb{R}^n$ is a vector space over the field $\mathbb{R}$ under our usual vector operations.
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Here's a quick review of these operations:
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\begin{itemize}
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\item Scalar multiplication is done elementwise: $3 \times [a, b, c] = [3a, 3b, 3c]$.
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\item Vector addition is similar: $[a, b, c] + [1, 2, 3] = [a+1,~b+2,~c+3]$.
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\item Vector addition is not valid for vectors of different sizes.
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\end{itemize}
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\problem{}
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Verify that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$ under these operations.
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\vfill
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\pagebreak
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We can also define an \textit{inner product} or \textit{vector product} that takes two elements of $V$ and produces another. \\
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When we work in $\mathbb{R}^n$, we usually use the dot product as our vector product. It is defined as follows: \\
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\definition{Dot Product}
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Given two vectors $a, b \in \mathbb{R}^n$, the \textit{dot product} of $a$ and $b$ (written $a \cdot b$ or $\langle a, b \rangle$) is $\sum_1^n a_ib_i$.
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\vspace{2mm}
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For example, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$,
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$$
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\langle a, b \rangle = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32
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$$
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As you may expect, the dot product $\langle a, b \rangle$ is valid iff $a$ and $b$ are the same size.
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\problem{}
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Show that the dot product is commutative.
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\vfill
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\problem{}
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Show that the dot product is positive-definite. \\
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This means that $\langle a, a \rangle > 0$ unless $a = 0$.
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\vfill
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\pagebreak
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