Alternating form
is multilinear and totally antisymmetric in its arguments.
Definition 3.33 (Alternating Form)
For vectors in or , the rank alternating form is defined by
Properties of alternating forms
Proposition 3.34 (Properties of Alternating Forms)
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is multilinear in its arguments. i.e.
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It is totally antisymmetric: for all .
Alternatively, for any permutation .
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.
Remark. Properties (1) (2) (3) uniquely define the alternating forms. Note that exchanging two vectors changes the sign of the alternating form, so if any two vectors are equal, the alternating form is zero.
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If for some , then . [Follows from (2).]
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If for some scalars , then . [Follows from (1) and (4).]
Nondegeneracy
Proposition 3.35
Proof of nondegeneracy
If the vectors are linearly dependent, then one of them can be written as a linear combination of the others, so the alternating form vanishes by multilinearity and antisymmetry.
Conversely, suppose are linearly independent. Then they span or , so for some matrix we can write . Hence, using antisymmetry,
Since , it follows that .
Related
Stated in
- Definition 3.33 (Alternating Form)§3.5.4 Alternating Forms
- Proposition 3.34 (Properties of Alternating Forms)§3.5.4 Alternating Forms
- Proposition 3.35§3.5.4 Alternating Forms