Rearranged Linear Maps

Advanced/Linear Maps/parts/2 matrices.tex

@@ -0,0 +1,34 @@
+\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