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R is a vector space

  1. Sep 22, 2014 #1
    Hi everyone!

    Can someone please tell me what is the proof of it? or Where can I find it?

    Thanks in advance.

    Best regards.
     
  2. jcsd
  3. Sep 22, 2014 #2
    Dear admin,

    Please remove/close this thread as I can prove it myself :)

    Thanks anyway.
     
  4. Sep 22, 2014 #3

    WWGD

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    Don't mean to nitpick , but it may be a good idea to specify the base field when you do these foundational exercises, i.e. the Reals are a vector space over 1-field? Clearly here there are not many options, but for R^2, the base field may be R or the Complexes (or maybe something else).
     
  5. Sep 23, 2014 #4
    Many thanks for the reply. Can you please tell me that what do you mean by specifying the base field?
     
  6. Sep 23, 2014 #5

    WWGD

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    Sure; when you say V is a vector space, you assume there is a set S of objects called vectors and a specific base field.
    For example, the set ## \mathbb R^2 ## seen as the vectors can be made into a vector space either over the field of Complex numbers,or over the set of Real numbers (or over any field F with ## \mathbb R \subset F \subset \mathbb C ##). Look at the 4 bottom axioms of vector spaces in e.g., http://en.wikipedia.org/wiki/Vector_space that specify how the field properties relate to the vectors and their resp. properties.
     
    Last edited: Sep 23, 2014
  7. Sep 24, 2014 #6

    HallsofIvy

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    The definition of "vector space" requires that it be "over" a given field. That is, one of the operations for a vector space is "scalar multiplication" in which a vector is multiplied by a member of the "base field". That field is typically the "rational numbers" (though rare), the "real numbers", or the "complex numbers".

    Any field can be thought of as a one-dimensional vector space over itself.
     
  8. Sep 24, 2014 #7

    WWGD

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    Ivy ,who/what are you replying to?
     
  9. Oct 3, 2014 #8

    HallsofIvy

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    I was responding to post #4 by WoundedTiger4: "Can you please tell me that what do you mean by specifying the base field?"
     
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