20 lines
1.3 KiB
TeX
20 lines
1.3 KiB
TeX
\section{Linear Independence}
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\begin{definition}[Linearly Dependent, Linearly Independent]
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A set $S$ of vectors is said to be \textit{linearly dependent} over a field $\F$ if there are vectors $v_1,v_2,\dots,v_n$ from $S$ and elements $a_1,a_2,\dots,a_n$ from $\F$, not all zero, such that $a_1v_1+a_2v_2+\dots+a_nv_n = 0$. A set of vectors that is not linearly dependent over $\F$ is called \textit{linearly independent} over $\F$.
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\end{definition}
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\begin{definition}[Basis]
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Let $V$ be a vector space over $\F$. A subset $B$ of $V$ is called a \textit{basis} for $V$ if $B$ is linearly independent over $\F$ and every element of $V$ is a linear combination of elements of $B$.
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\end{definition}
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\begin{theorem}[Invariance of Basis Size]
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If $\{u_1,u_2,\dots,u_m\}$ and $\{w_1,w_2,\dots,w_n\}$ are both bases of a vector space $V$ over a field $\F$, then $m=n$.
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\end{theorem}
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\begin{definition}[Dimension]
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A vector space that has a basis consisting of $n$ elements is said to have \textit{dimension $n$}. For completeness, the trivial vector space $\{0\}$ is said to be spanned by the empty set and to have dimension 0.
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\noindent A vector space that has a finite basis is called \textit{finite dimensional}; otherwise, it is called \textit{infinite dimensional}.
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\end{definition}
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