dual basis in a sentence

Examples

1. There are as many dual basis vectors \ omega ^ i as there are basis vectors \ mathbf { e } _ i, so the dual space has the same dimension as the linear space itself.
2. Using an explicit basis for " V " and the corresponding dual basis for " V "  ", one can show that this gives the same definition of the trace as given above.
3. The association of a dual basis with a basis gives a map from the space of bases of " V " to the space of bases of " V "  ", and this is also an isomorphism.
4. A basis can be chosen according to such a form, and actually that's often what one does; e . g ., orthonormal basis . ( I just noticed the article doesn't even mention dual basis, which is an important concept .)
5. Given a nondegenerate quadratic form on " V ", " V "  " becomes naturally identified with " V ", and the dual basis may be regarded as elements of " V ", but are not in general the same set as the original basis.
6. T1 follows the G3 ( MP2 ) recipe, however, by substituting an HF / 6-31G * for the MP2 / 6-31G * geometry, eliminating both the HF / 6-31G * frequency and QCISD ( T ) / 6-31G * energy and approximating the MP2 / G3MP2large energy using dual basis set RI-MP2 techniques, the T1 method reduces computation time by up to 3 orders of magnitude.