Understanding the Basis of a Zero Vector Space

[mathmathmad]

Homework Statement

erm, I just want to know, what is the basis for a zero vector space?

Homework Equations

The Attempt at a Solution

is it the zero vector itself? but if that's the case, then the constant alpha could be anything other than zero, which means the zero vector is not linearly independent...


another quick question, how do you determine the basis of a polynomial vector space in general?

 

[radou]

A basis is usually not defined for the trivial vector space, e.g. the basis is the empty set.

 

[mathmathmad]

does that mean the basis for zero vector space is just ∅?

another quick question, how do you determine the basis of a polynomial vector space in general?

 

[radou]

Unless I'm mistaken, yes. Although, some confirmation from the pro mathematicians here would be great.

 

[HallsofIvy]

Yes, the vector space consisting only of the 0 vector has the empty set as basis.

There is no such thing as "the" basis of any vector space. Except for the trivial case above, any vector space has an infinite number of possible bases. The "standard" basis for the vector space of polynomials of dimension n or less is {1, x, x², ..., xⁿ}. If you are talking about finding a basis for some subspace of such a space, how you would do that depends upon how the subspace is defined.