Linear independence and dependence
forces for every : the vectors are linearly independent; otherwise they are linearly dependent.
Characterisations
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A set of vectors is linearly dependent if and only if one of the vectors can be expressed as a linear combination of the others.
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In , vectors are linearly independent iff : geometrically they are not coplanar, the left-hand side being the volume of the parallelepiped they span.
Examples
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in is linearly dependent, since .
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in is linearly independent.
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Any set containing is linearly dependent.
Related
Stated in
- Definition 2.50 (Linear Independence and Dependence)§2.10.1 Linear Independence and Dependence