Talk:PlanetPhysics/Vector Space 2
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[edit source]%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: vector space %%% Primary Category Code: 02.10.Ud %%% Filename: VectorSpace2.tex %%% Version: 1 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
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Let $F$ be a \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} (or, more generally, a division ring). A \emph{vector space} $V$ over $F$ is a set with two \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html}, $+: V \times V \longrightarrow V$ and $\cdot: F \times V \longrightarrow V$, such that
\begin{enumerate} \item $(\u+\v)+\w = \u+(\v+\w)$ for all $\u,\v,\w \in V$ \item $\u+\v=\v+\u$ for all $\u,\v\in V$ \item There exists an element $\0 \in V$ such that $\u+\0=\u$ for all $\u \in V$ \item For any $\u \in V$, there exists an element $\v \in V$ such that $\u+\v=\0$ \item $a \cdot (b \cdot \u) = (a \cdot b) \cdot \u$ for all $a,b \in F$ and $\u \in V$ \item $1 \cdot \u = \u$ for all $\u \in V$ \item $a \cdot (\u+\v) = (a \cdot \u) + (a \cdot \v)$ for all $a \in F$ and $\u,\v \in V$ \item $(a+b) \cdot \u = (a \cdot \u) + (b \cdot \u)$ for all $a,b \in F$ and $\u \in V$ \end{enumerate}
Equivalently, a vector space is a \htmladdnormallink{module}{http://planetphysics.us/encyclopedia/RModule.html} $V$ over a ring $F$ which is a field (or, more generally, a division ring).
The elements of $V$ are called \emph{\htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html}}, and the element $\0 \in V$ is called the \emph{zero vector} of $V$.
This entry is a copy of the GNU FDL vector space article from \htmladdnormallink{PlanetMath}{http://planetmath.org/encyclopedia/VectorSpace.html}. Author of the original article: djao. History page of the original is \htmladdnormallink{here}{http://planetmath.org/?op=vbrowser&from=objects&id=364}
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