Add theorem commands

MA2008/linear-transforms.tex

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+\input{decls.tex}
+\title{Vector Spaces and Linear Transformations}
+\begin{document}
+\maketitle
+
+\begin{description}
+\item[Linear Transformation] A function $\phi: V \to W$ between vector spaces $V$ and $W$ (over some field $K$), such that
+	\begin{align*}
+		\phi(v + w) & \equiv \phi(v) + \phi(w) \\
+		\phi(x \cdot v) & \equiv x \cdot \phi(v) \tag{For $x \in K$}
+	\end{align*}
+\end{description}
+
+Differentiation is a linear transformation.
+Solutions to $f'' + f = 0$ for function $f$ are a vector space.
+
+\begin{theorem}
+For any scalars $\lambda, \mu \in \R$, there is a unique solution such that $f(0) = \mu$ and $f'(0) = \lambda$
+\end{theorem}
+The vector space is then two-dimensional, with basis $sin(x), cos(x)$
+
+
+\subsection*{}
+Vector spaces are used over finite fields in \emph{Algebraic Coding Theory}. The field is $\mathbb{F}_2 = \{0, 1\}$ - the integers mod 2.
+Binary strings of length $n$ are then a vector space over $\mathbb{F}_2^n$.
+ECC can be based on vector subspaces of $F_2^n$. (Vector subspaces are closed subsets of a vector space).
+\end{document}