| Version 23 |
Version 22 |
| Let $V$ be a vector space over a |
Let $V$ be a vector space over a |
| field $F$. We say that $v_1,\ldots, v_k\in V$ are linearly dependent if there exist scalars $\lambda_1,\ldots, \lambda_k\in F$, not all zero, such that |
field $F$. Then for scalars $\lambda_1,~ \lambda_2,~ \ldots, ~\lambda_n \in F$ the vectors $\vec{v}_1,~ \vec{v}_2,~ \ldots,~ \vec{v}_n \in V$ are |
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linearly independent if the following condtion holds: |
| \[ |
\[ |
| \lambda_1 v_1+ \cdots +\lambda_k v_k = 0 . |
\lambda_1\vec{v}_1+ \lambda_2 \vec{v}_2 + ~\cdots ~ +\lambda_n \vec{v}_n = 0 \mbox{ implies } |
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~ \lambda_1 = \lambda_2 = ~\ldots~ = \lambda_n=0 |
| \] |
\] |
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If no such scalars exist, then we say that the vectors are \emph{linearly independent}.
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Otherwise, if this conditions fails, the vectors are said to be linearly dependent. Furthermore, an infinite set of vectors is linearly independent if all \emph{finite} subsets are linearly
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| More generally, we say that a (possibly infinite) subset $S\subset V$ is linearly independent if all finite subsets of $S$ are linearly independent. |
independent. |
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| In the case of two vectors, linear dependence means that one of the |
In the case of two vectors, linear independence means that one of these vectors is not a scalar multiple of the other. |
| vectors is a scalar multiple of the other. As an alternate |
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| characterization of dependence, we also have the following. |
As an alternate characterization of dependence, we have that a set of of vectors is linearly dependent if and only if some vector in the |
| \begin{proposition} |
set lies in the linear span of the other vectors in the set. |
| Let $S\subset V$ be a subset of a vector space. Then, $S$ is |
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| linearly dependent if and only if there exists a $v\in S$ such that |
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| $v$ can be expressed as a linear combination of the vectors in the |
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| set $S\backslash \{v\}$ (\PMlinkname{all the vectors in $S$ other |
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| than $v$}{SetDifference}). |
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| \end{proposition} |
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