\section{Linearity}

\definition{}
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 $f: A \to B$.


\definition{}
Let $V$ and $U$ be vector spaces, and let $f: V \to U$ be a map from $V$ to $U$. \\
We say $f$ is \textit{linear} if it satisfies the following for any $v \in V$, $u \in U$, $a \in \mathbb{R}$:
\begin{itemize}
	\item $f(u + v) = f(u) + f(v)$
	\item $f(au) = af(u)$
\end{itemize}
In other words, $f$ is linear if it is \say{closed} under addition and scalar multiplication.


\problem{}
It is often convenient to combine the two conditions above into one. \\
Show that $f(au + v) = af(u) + f(v)$ iff $f$ is linear.

\vfill

\problem{}
Is $f(x) = mx + b$ a linear map on $\mathbb{R}$?

\vfill
\problem{}
In general, what does a linear map in $\mathbb{R}^n$ look like?

\vfill

\problem{}
Is $\text{median}(v): \mathbb{R}^n \to \mathbb{R}$ a linear map on $\mathbb{R}^n$?


\vfill

\problem{}
Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$?

\vfill
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