\section{Matrices}

\theorem{}<thebigtheorem>
Any linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as an $n \times m$ matrix. \\
Conversely, every $n \times m$ matrix represents a linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ \\

\vspace{2mm}

In other words, \textbf{matrices are linear transformations}. \\
The next two problems provide a proof.


\problem{}<prooffwd>
Let $A$ be an $m \times n$ matrix, and $v$ an $m \times 1$ vector. \\
Show that the map $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\



\vfill

\problem{}<proofback>
Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$.


\vfill
\pagebreak

\problem{}
Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}(p)$. \\
Find a matrix that corresponds to $D$. \\
\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How so?}

\vfill
\pagebreak