Proof of R being a Vector Space - Best Resources Available

In summary, the conversation is about specifying the base field in exercises involving vector spaces. The original poster asks for clarification on what is meant by specifying the base field, and another user provides an explanation using the example of ##\mathbb{R}^2## being a vector space over either the set of real numbers or the set of complex numbers. The conversation also touches on the definition of a vector space and the role of the base field in scalar multiplication.
  • #1
woundedtiger4
188
0
Hi everyone!

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

Thanks in advance.

Best regards.
 
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  • #2
Dear admin,

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

Thanks anyway.
 
  • #3
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
WWGD said:
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
woundedtiger4 said:
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
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
Ivy ,who/what are you replying to?
 
  • #8
I was responding to post #4 by WoundedTiger4: "Can you please tell me that what do you mean by specifying the base field?"
 

Related to Proof of R being a Vector Space - Best Resources Available

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects, called vectors, and two operations, vector addition and scalar multiplication, that satisfy certain properties. Essentially, a vector space is a way to represent and manipulate mathematical objects in a systematic way.

2. What is the importance of proving that R is a vector space?

Proving that R, or the set of real numbers, is a vector space is important because it helps us understand the properties and behaviors of real numbers in a more structured and organized manner. It also allows us to use the powerful tools and techniques of vector spaces to study and solve problems related to real numbers.

3. What are the main properties that need to be satisfied for R to be considered a vector space?

There are 10 main properties, or axioms, that need to be satisfied for R to be considered a vector space. These include closure under addition and scalar multiplication, commutativity and associativity of addition, existence of additive and multiplicative identities, and distributivity of scalar multiplication over vector addition.

4. What are some good resources for learning about the proof of R being a vector space?

Some good resources for learning about the proof of R being a vector space include textbooks on linear algebra or abstract algebra, online lecture notes from universities, and videos from educational websites such as Khan Academy or MIT OpenCourseWare.

5. How can understanding vector spaces help in other areas of science?

Understanding vector spaces can be helpful in many areas of science, such as physics, engineering, and computer science. Vector spaces provide a framework for understanding and analyzing multi-dimensional systems and their transformations. They can also be used to model and solve problems related to linear systems, such as equations of motion or electrical circuits.

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