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
Famous quotes containing the words higher and/or dimensions:
“Our first line of defense in raising children with values is modeling good behavior ourselves. This is critical. How will our kids learn tolerance for others if our hearts are filled with hate? Learn compassion if we are indifferent? Perceive academics as important if soccer practice is a higher priority than homework?”
—Fred G. Gosman (20th century)
“Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
—bell hooks (b. c. 1955)