Relationship With Other Formalisms
may be directly compared to vector algebra.
The even subalgebra of is isomorphic to the complex numbers, as may be seen by writing a vector P in terms of its components in an orthonormal basis and left multiplying by the basis vector e1, yielding
where we identify i ↦ e1e2 since
Similarly, the even subalgebra of with basis {1, e2e3, e3e1, e1e2} is isomorphic to the quaternions as may be seen by identifying i ↦ −e2e3, j ↦ −e3e1 and k ↦ −e1e2.
Every associative algebra has a matrix representation; the Pauli matrices are a representation of and the Dirac matrices are a representation of, showing the equivalence with matrix representations used by physicists.
Read more about this topic: Geometric Algebra
Famous quotes containing the words relationship with and/or relationship:
“Every man is in a state of conflict, owing to his attempt to reconcile himself and his relationship with life to his conception of harmony. This conflict makes his soul a battlefield, where the forces that wish this reconciliation fight those that do not and reject the alternative solutions they offer. Works of art are attempts to fight out this conflict in the imaginative world.”
—Rebecca West (1892–1983)
“But the relationship of morality and power is a very subtle one. Because ultimately power without morality is no longer power.”
—James Baldwin (1924–1987)