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Linear Space

  1. Feb 4, 2009 #1
    I was wondering, some of the things that define a Linear Space such as:

    [tex]v \in V[/tex] then [tex]1v = v[/tex] or [tex]\vec{0} \in V[/tex] such that [tex]\vec{0} + v = v[/tex]

    They seem very obvious and intuitive, but, is there ever a time they break down in the Real plane? I think they might break down in the complex plane, but, I'm not too sure how they would.
  2. jcsd
  3. Feb 4, 2009 #2
    What do you mean, break down?
  4. Feb 4, 2009 #3
    The two pretty obvious & intuitive properties. Where they don't work anymore; such that when you have v living in V and you multiply 1 by v, it longer equals v or add the zero vector it doesn't equal itself?
  5. Feb 4, 2009 #4
    Then you don't have a vector space.
  6. Feb 4, 2009 #5
    Exactly, but, I'm wondering when does this definition not hold true. These two properties seem pretty obvious, and pretty intuitive. More than anything, I'm wondering why are they included when defining a Linear Space. Other than for extra-proofing.

    And if these definitions fail.
  7. Feb 5, 2009 #6
    They are included because they are useful; there are many interesting theorems about vector spaces, and there are lots of things that can be modeled by vector spaces. If you omit some axioms such as 1v = v or 0 + v = v, then many of the theorems fail to be true.
  8. Feb 5, 2009 #7


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    They seem pretty intuitive, because every object that has been introduced to you as a linear space has those properties. If you have a set with an operation, we almost always use 0 and 1 to be defined as the additive and multiplicative identities; so if you had a set with an operation that had no such identity, we wouldn't call elements 0 and 1.

    Off the top of my head I'm not able to think of a space that satisfies every vector space axiom except for those two.
  9. Feb 12, 2009 #8
    The statements 0+v= v and 1v=v are really saying that there are additive and multiplicative identities and telling you particularly what they are (0 & 1).
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