Proving V is a Real Vector Space Given V is a Complex Vector Space

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Homework Help Overview

The discussion revolves around proving that if V is a complex vector space, then V is also a real vector space. The original poster expresses difficulty in constructing a proof and is seeking guidance on how to approach the problem using the definitions of vector spaces and relevant axioms.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need to prove the properties of a real vector space based on the definitions of complex and real vector spaces. There are attempts to clarify the requirements for a direct proof and the necessity of defining a new scalar multiplication operation for real numbers.

Discussion Status

The conversation is ongoing, with participants providing insights into the proof structure and emphasizing the importance of defining scalar multiplication appropriately. There is no explicit consensus yet, but productive suggestions have been made regarding the proof approach.

Contextual Notes

Participants note the need to establish a new scalar multiplication operation to transition from complex to real scalars, which is a critical step in the proof process.

bendaddy
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Homework Statement


Hello, I'm having a little difficulty with this proof.
Prove: If V is a complex vector space, then V is also a real vector space.


Homework Equations


Definition 1: A vector space V is called a real vector space if the scalars are real numbers.
Definition 2: A vector space V is called a complex vector space if the scalars are complex numbers.


The Attempt at a Solution


I've tried a proof by contradiction saying that "V is a complex vector space and V is not a real vector space." Can't seem to find any sort of contradiction throughout my argument though.
 
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It's good that you didn't find any contradictions, because there aren't any. The set of vectors in a complex vector space with complex scalars are also a vector space over the reals. Just try to prove it directly. What things do you need to prove?
 
I just need to develop a proof using the vector space axioms and the two definitions listed above that if V is a complex vector space, then V is also a real vector space. Not sure how to go about proving it directly though...
 
bendaddy said:
I just need to develop a proof using the vector space axioms and the two definitions listed above that if V is a complex vector space, then V is also a real vector space. Not sure how to go about proving it directly though...

State the axioms you have to prove. Take them one by one.
 
Before you can check any of the vector space axioms, you must define a new scalar multiplication operation. The one you have is a map from ℂ×V into V. You need to use it to define a map from ℝ×V into V.
 

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