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Complex/Real Vector Spaces

  1. Mar 23, 2012 #1
    1. The problem statement, all variables and given/known data
    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.


    2. Relevant 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.


    3. 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.
     
  2. jcsd
  3. Mar 23, 2012 #2

    Dick

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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?
     
  4. Mar 23, 2012 #3
    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....
     
  5. Mar 23, 2012 #4

    Dick

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    State the axioms you have to prove. Take them one by one.
     
  6. Mar 24, 2012 #5

    Fredrik

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    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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