# R is a vector space

Hi everyone!

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

Best regards.

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Thanks anyway.

WWGD
Gold Member
2019 Award
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).

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
Gold Member
2019 Award
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.

Last edited:
HallsofIvy
Homework Helper
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