Here | v \ rangle is the standard basis element of \ mathbf { R } ^ n with a one in the v ^ { th } position.
22.
Often the basis elements " Q " come in complex conjugate pairs, so the real subspace can be recovered as the fixed points of complex conjugation.
23.
Insofar as \ mathfrak { g } is just an ordinary vector space, the generators of \ mathfrak { g } correspond to basis elements of the vector space.
24.
In contrast, non-zero free abelian groups are never divisible, because it is impossible for any of their basis elements to be nontrivial integer multiples of other elements.
25.
Green's functions can be expanded in terms of the basis elements ( harmonic functions ) which are determined using the separable coordinate systems for the linear partial differential equation.
26.
From the PBW theorem, it is clear that all such central elements will be linear combinations of symmetric homogenous polynomials in the basis elements e _ a of the Lie algebra.
27.
For each value of " k " there are " n " choose " k " basis elements, so the total dimension of the Clifford algebra is
28.
But in 3D, with three vector directions, there are three bivector basis elements ? 1 ? 2, ? 2 ? 3, ? 3 ? 1, so three imaginaries.
29.
You can send the basis elements to their new spots with reflections one at a time and show that the later reflections don't mess with the basis elements that were already set.
30.
You can send the basis elements to their new spots with reflections one at a time and show that the later reflections don't mess with the basis elements that were already set.
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