16 lines
934 B
TeX
16 lines
934 B
TeX
\section{Definition and Examples}
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\begin{definition}[Vector Space]
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A set $V$ is said to be a \textit{vector space} over a field $\F$ if $V$ is an Abelian group under addition (denoted by $+$) and, if for each $a \in \F$ and $v \in V$, there is an element $av \in V$ such that the following conditions hold for all $a,b \in \F$ and all $u,v \in V$.
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\begin{enumerate}
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\item $a(v + u) = av + au$
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\item $(a + b)v = av + bv$
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\item $a(bv)=(ab)v$
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\item $1v=v$
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\end{enumerate}
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\end{definition}
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\begin{remark}
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The members of a vector space are called \textit{vectors}. The members of the field are called \textit{scalars}. The operation that combines a scalar $a$ and a vector $v$ to form the vector $av$ is called \textit{scalar multiplication}. In general, we will denote vectors by letters from the end of the alphabet, such as $u,v,w$, and scalars by letters from the beginning of the alphabet, such as $a,b,c$.
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\end{remark}
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