# Basis of or basis for in mathematics

I read this stackexchange about whether to say "basis of" or "basis for". Does the answer given there (that both are correct, of is newer and for is still more common) apply to mathematics as well?

• What basis are we talking about, exactly? The word basis does behave exactly the same in mathematics as everywhere else, but I have that nagging suspicion you're really asking about the word base. (For which other languages also use basis, but English does not, so this question might be the result of a poor false-friend translation.) For example, the base of a logarithm is always the base of it, never the base for it. And again, never its basis. So please elaborate what it is that you're after, and please supply the exact example sentence(s) you are looking at right now. Jul 5 '18 at 13:25
• I'm not asking about base, I'm asking about basis as in the basis of a vector space. The word does not behave completely the same as outside mathematics, as "Forming the basis for could mean it is one of the factors forming the basis" cannot be the case in mathematics. A set either is or is not a basis for some space. Jul 5 '18 at 13:28
• That was a quote from the accepted answer of the stackexchange question about basis for/basis of, by the way Jul 5 '18 at 13:29
• An example sentence would therefore be "Let B be a basis for/of V". Jul 5 '18 at 13:33
• I don't agree with the answer you mention; 'the' not 'a' is used. Though I'd say there's not a great deal of difference as regards choice of prepositions, 'of' connotes the finished structure and 'for' the actual building of the structure more strongly. Jul 5 '18 at 13:47

The two forms are completely interchangeable. English in mathematical publications is "Global English", and often written by non-native speakers who get the traditional grammar wrong, but it doesn't matter because the meaning is carried by the mathematical symbols.

As I expect you know, for and of have directional connotations, with for meaning towards and of meaning away from. However, the association between a vector space and a basis works both ways.

That said, your reference to the basis and the vector space may be written in a directional context, e.g. with one step preceding another in your exposition. One form may sound more natural than the other.

Or you may have a personal preference based on how you see a basis being used in general, e.g. you find a basis because you intend to use it for something. However, this varies according to a person's first language, which basically gets us back to Global English and the interchangeability of the two forms.

I think the prepositions are (as they are in English, in natural language):

Representation of

The latter might be what has you confused. When talking about V, you might say B represents a basis. Specifically, it is a basis (of / for) V. So in your example sentence, I would write:

B is a basis for V

But you could equivalently say:

a basis representation of vector space V

Whether you use of or for may depend on dialect or style. Just like some use or omit the definite article in the following sentence:

going to (the) hospital

I also looked at the Ngram comparing be a basis of and be a basis for, as that construction with be seems common in science (and less so in natural language). It shows the version with for is more common. The trends seems to be the same when changing to American or British English.

• They're both used, and this Ngram seems to indicate they're used with roughly equal frequency. Your Ngram has lots of hits like be a basis for subcategorization/comparison/authentic faith, which might distort the results. Jul 5 '18 at 13:58
• All right, so to avoid that I looked up orthonormal basis for/of and then it looks like they are really completely interchangeable. I wonder if it has any influence if there are multiple bases (or spaces) at play.. Jul 5 '18 at 14:06