Proof of R being a Vector Space - Best Resources Available

[woundedtiger4]

Hi everyone!

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

Thanks in advance.

Best regards.

 

[woundedtiger4]

Dear admin,

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

Thanks anyway.

 

[WWGD]

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).

 

[woundedtiger4]

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?

 

[WWGD]

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.

 

[HallsofIvy]

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.

 

[WWGD]

Ivy ,who/what are you replying to?

 

[HallsofIvy]

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