annihilator of vector subspace
If is a vector space, and is any subset of , the annihilator of , denoted by , is the subspace of the dual space that kills every vector in :
Similarly, if is any subset of , the annihilated subspace of is
(Note: this may not be the standard notation.)
1 Properties
Assume is finite-dimensional. Let and denote subspaces of and , respectively, and let denote the natural isomorphism from to its double dual .
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i.
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ii.
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iii.
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iv.
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v.
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vi.
(a dimension theorem)
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vii.
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viii.
, where denotes the sum of two subspaces of .
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ix.
If is a linear operator, and , then the image of the pullback is .
References
- 1 Friedberg, Insel, Spence. Linear Algebra. Prentice-Hall, 1997.
Title | annihilator of vector subspace |
---|---|
Canonical name | AnnihilatorOfVectorSubspace |
Date of creation | 2013-03-22 15:25:59 |
Last modified on | 2013-03-22 15:25:59 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 5 |
Author | stevecheng (10074) |
Entry type | Definition |
Classification | msc 15A03 |
Defines | annihilator |
Defines | annihilated subspace |