Why is the zero element unique if cos(0) = cos(2π)?

[randommacuser]

Suppose I have a set involving trigonometric functions, with addition defined as multiplication of two vectors. If this is a vector space, the zero vector has to be unique. If cos (0) works as the zero vector, then cos (2*pi), etc. also work. Does this mean the set is not a vector space, because the zero element is not unique? Or is it still a vector space (all other axioms check out) because cos (0) = cos (2*pi) = 1 ?

 

[snoble]

I'm a little confused... what's your underlying field? How is multiplication between a field element and vector element defined?

Steven

 

[jcsd]

As long as it satifies the axioms of a vector space it is a vector space. The set of vectors may be a partition of another set defined by an equiavlence relation (which I think is what you're getting at).

 

[matt grime]

The zero vector in a vector space of functions is the ZERO function if f(x)=0 for all x. Is that the kind of thing you're after?

 

[randommacuser]

I understand now. You guys were a lot of help!