\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}
