basis

noun

1. A physical base or foundation.

   “1695, William Congreve, To the King, on the taking of Namur, 1810, Samuel Johnson, Alexander Chalmers (biographies), The Works of the English Poets from Chaucer to Cowper, Volume 10, page 271, Beholding rocks from their firm basis rent; Mountain on mountain thrown, With threatening hurl, that shook th' aerial firmament!”

   “We see here the ground-plan of masses of houses, with their upper walls of fire-baked brick on a basis of stone.”

2. A starting point, base or foundation for an argument or hypothesis.

   “I wonder if the South Korean side has any basis that its smog is from China.”

3. An underlying condition or circumstance.

   “Hodgson may now have to bring in James Milner on the left and, on that basis, a certain amount of gloss was taken off a night on which Welbeck scored twice but barely celebrated either before leaving the pitch angrily complaining to the Slovakian referee.”

4. A regular frequency.

   “You should brush your teeth on a daily basis at minimum.”

   “The flights to Fiji leave on a weekly basis.”

   “Cars must be checked on a yearly basis.”

5. The difference between the cash price a dealer pays to a farmer for his produce and an agreed reference price, which is usually the futures price at which the given crop is trading at a commodity exchange.

   “Included in the basis could be elevation, cleaning, freight by truck and/or rail, government inspection fees, administration fees, interest and storage charges as well as allowance for risk and profit for the grain dealer.”

6. In a vector space, a linearly independent set of vectors spanning the whole vector space.
7. Amount paid for an investment, including commissions and other expenses.
8. A collection of subsets ("basis elements") of a set, such that this collection covers the set, and for any two basis elements which both contain an element of the set, there is a third basis element contained in the intersection of the first two, which also contains that element.

   “The collection of all possible unions of basis elements of a basis is said to be the topology generated by that basis.”