Rearranged handout

Advanced/Linear Maps/parts/2 linear.tex

@@ -0,0 +1,43 @@
+\section{Linear Transformations}
+
+\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 $\mathbb{P}^n$? \\
+\hint{$\mathbb{P}^n$ is the set of all polynomials of degree $n$.}
+
+\vfill
+\pagebreak