Higher Dimensions
An n-dimensional Euclidean space allows the existence of multivectors of grade n (n-vectors). The dimension of the vector space is evidently equal to n and a simple combinatorial analysis shows that the dimension of the bivector space is . In general, the dimension of the multivector space of grade m is and the dimension of the whole Clifford algebra is .
A given multivector with homogeneous grade is either invariant or changes sign under the action of the reversion conjugation . The elements that remain invariant are defined as Hermitian and those who change sign are defined as anti-Hermitian. The diverse grades can be classified accordingly, as shown in the next table
| Grade | Classification |
|---|---|
| Hermitian | |
| Hermitian | |
| Anti-Hermitian | |
| Anti-Hermitian | |
| Hermitian | |
| Hermitian | |
| Anti-Hermitian | |
| Anti-Hermitian | |
Read more about this topic: Paravector
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