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Vector spaces

  1. Oct 17, 2009 #1
    Hello, I'm studying linear algebra and wanted to know what is the difference between a "vector space" and a "vector space over field F".
    I know that a vector space over field F satisfies the 8 axioms, but does a vector space satisfy this also?
     
  2. jcsd
  3. Oct 17, 2009 #2
    A field could be a set of real, or complex numbers, or a set of rational numbers whose addition and multiplication is well defined.
     
  4. Oct 17, 2009 #3
    This doesn't answer my question, I want to know the difference between a vector space and a vector space over field F.
     
  5. Oct 17, 2009 #4

    Fredrik

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    There's no difference. Every vector space is a vector space over a field. The field is usually the real numbers or the complex numbers, but it could be any field. A vector space over [itex]\mathbb R[/itex] is often called a "real vector space", and a vector space over [itex]\mathbb C[/itex] is often called a "complex vector space".
     
  6. Oct 17, 2009 #5
    Right this would make sense. So that means the 8 axioms for a "Vector Space over a field F" also hold for a "Vector space".

    Because i know that the basic defin of a vector space is that:
    1) It contains a non empty set V whose elements are vectors
    2) A field F whose elements are scalars
    3) A binary operation + on V Under which V is closed
    4) A multiplication . of a vector by a scalar.
    So on top of this the 8 axioms (That hold for a vector space over a field F) also hold for a vector space (i cant be asked to write all axioms)?
     
  7. Oct 17, 2009 #6

    Fredrik

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    Yes, you got it right. I prefer to state the definition a bit differently though. In my definition, V is a set, [itex]\mathbb F[/itex] is a field, and [itex]A:V\times V\rightarrow V[/itex] and [itex]S:\mathbb F\times V\rightarrow V[/itex] are functions (called "addition" and "scalar multiplication" respectively). We use the notation [itex]A(x,y)=x+y[/itex] and [itex]S(k,x)=kx[/itex].

    Definition: A 4-tuple [itex](V,\mathbb F,A,S)[/itex] is said to be a vector space over the field [itex]\mathbb F[/itex] if

    (i) [itex](x+y)+z=x+(y+z)[/itex] for all [itex]x,y,z\in V[/itex]

    ...and so on. (You seem to know the rest).

    Note that V is just a set. It's convenient to call V a vector space, but you should be aware that this is actually a bit sloppy. It's certainly OK to do it when it's clear from the context what field [itex]\mathbb F[/itex] and what addition and scalar multiplication functions we have in mind. For example, it's common to refer to "the vector space [itex]\mathbb R^2[/itex] " because everyone is familiar with the standard vector space structure on that set.
     
  8. Oct 17, 2009 #7
    I find it useful to think of vector spaces as special cases of modules over rings.
    A module is just an abelian group together with a distributive multiplication by elements of a ring. If the ring is a field then the module is a vector space.

    The distinguishing feature of a field is that is has multiplicative inverses.

    Much of the theory of vector spaces actually comes from considering modules where the ring is a principal ideal domain. This is because the ring of polynomials over a field is a principal ideal domain.
     
    Last edited: Oct 17, 2009
  9. Oct 18, 2009 #8
    right thanks fredrik
     
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