58 lines
1.5 KiB
TeX
58 lines
1.5 KiB
TeX
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\section{Linear Maps}
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\definition{}
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A \textit{function} or \textit{map} $f$ from a set $A$ to a set $B$ is a rule that assigns an element of $B$ to each element of $A$. We write this as $f: A \to B$.
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\definition{}<lineardef>
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Let $f: U \to V$ be a map. \\
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We say $f$ is \textit{linear} if it satisfies the following for any $u \in U$, $v \in V$, $a \in \mathbb{R}$:
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\begin{itemize}
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\item $f(u + v) = f(u) + f(v)$
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\item $f(au) = af(u)$
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\end{itemize}
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In other words, $f$ is linear if it is \say{closed} under addition and scalar multiplication.
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\problem{}
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It is often convenient to combine the two conditions above into one. \\
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Show that $f(au + v) = af(u) + f(v)$ iff $f$ is linear. Use \ref{lineardef}.
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\vfill
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\problem{}
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Is $f(x) = mx + b$ a linear map?
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\vfill
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\problem{}
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In general, what does a linear map in $\mathbb{R} \to \mathbb{R}$ look like?
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\vfill
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\pagebreak
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\problem{}
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Is the map ${median}(v): \mathbb{R}^3 \to \mathbb{R}$ linear? \\
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\hint{$median([3, 5, 4]) = 4$, but you already knew that.}
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\vfill
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\problem{}
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Is the map $f(v): \mathbb{R}^3 \to \mathbb{R}$ defined by $f(v) = v_0 + 2v_1 + v_2$ linear? \\
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\hint{$v_n$ is the $n^\text{th}$ element of $v$. $v$ is a 3-element vector.}
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\vfill
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\problem{}
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Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? \\
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\vspace{1mm}
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\hint{$\mathbb{P}^n$ is the set of polynomials with degree at most $n$.}
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\vfill
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\problem{}
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In general, what does a linear map from $\mathbb{R}^m \to \mathbb{R}^n$ look like?
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\vfill
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\pagebreak
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