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:
“Man is a stream whose source is hidden. Our being is descending into us from we know not whence. The most exact calculator has no prescience that somewhat incalculable may not balk the very next moment. I am constrained every moment to acknowledge a higher origin for events than the will I call mine.”
—Ralph Waldo Emerson (1803–1882)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)