Question about zero element in vector spaces

In summary, the conversation discusses whether a set involving trigonometric functions is a vector space based on its definition of addition and the uniqueness of the zero vector. It is determined that as long as the set satisfies the axioms of a vector space, it can still be considered a vector space even if the zero vector is not unique. The concept of a partition and equivalence relation is also mentioned. The conversation concludes with a clarification on the definition of the zero vector in a vector space of functions.
  • #1
randommacuser
24
0
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 ?
 
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  • #2
I'm a little confused... what's your underlying field? How is multiplication between a field element and vector element defined?

Steven
 
  • #3
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).
 
  • #4
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?
 
  • #5
I understand now. You guys were a lot of help!
 

What is the zero element in a vector space?

The zero element, also known as the zero vector, is a special vector in a vector space that has a magnitude of 0 and a direction that is undefined. It is often denoted as ∅ or 0 and is the additive identity element in a vector space.

Why is the zero element important in vector spaces?

The zero element is important because it allows for the existence of the additive inverse of any vector in a vector space. This means that for any vector v in a vector space, there exists a vector -v which when added to v results in the zero element.

Can a vector space have more than one zero element?

No, a vector space can only have one zero element. This is because the zero element must satisfy certain properties, such as being the additive identity and having a magnitude of 0, which can only be fulfilled by one unique vector.

What happens when a vector is multiplied by the zero element?

When a vector is multiplied by the zero element, the resulting vector is also the zero element. This is because the scalar multiplication of 0 by any vector results in the zero vector.

Is the zero element considered a basis vector in a vector space?

No, the zero element is not considered a basis vector in a vector space. A basis vector must be linearly independent and the zero element is a linear combination of itself, making it a linearly dependent vector.

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