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What is a basis?

What is a basis? Here are some definitions.

Noun
  1. A physical base or foundation.
  2. A starting point, base or foundation for an argument or hypothesis.
  3. An underlying condition or circumstance.
  4. A regular frequency.
  5. (linear algebra) In a vector space, a linearly independent set of vectors spanning the whole vector space.
  6. (accounting) Amount paid for an investment, including commissions and other expenses.
  7. (topology) 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.
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