R is a vector space

  • #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.
 

Answers and Replies

  • #2
Dear admin,

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

Thanks anyway.
 
  • #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).
 
  • #4
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).
Many thanks for the reply. Can you please tell me that what do you mean by specifying the base field?
 
  • #5
WWGD
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Many thanks for the reply. Can you please tell me that what do you mean by specifying the base field?
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.
 
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  • #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.
 
  • #7
WWGD
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Ivy ,who/what are you replying to?
 
  • #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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