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Homework Help: Linear Algebra - Vector Spaces

  1. Sep 28, 2005 #1
    Hi,
    I'm having trouble with these homework questions.

    I have to prove that B*0v = 0v , where B is a scalar.

    Also, I have to prove that if aX = 0v , then either a = 0 or X = 0
    ---where a is a scalar and X is a vector.


    I know that I have to use the 8 axioms but I'm not sure where to begin.
     
  2. jcsd
  3. Sep 28, 2005 #2
    One of the axioms is that anything times a 0 vector is 0, unless that is the one you are trying to prove.

    The zero vector will contain all zero components, therefore any scalar multiplied by 0 will also be 0 as 0*a is always zero for any scalar. For your first question, the proof depends on what "B" represents.
     
  4. Sep 28, 2005 #3
    Thanks for answering the first question, but any ideas about the second (aX = 0)? These vector spaces really have me confused.
     
  5. Sep 28, 2005 #4

    HallsofIvy

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    Science Advisor

    No, "0 times anything is 0" is never an axiom- it's too easy to prove!
    And I don't like the idea of using components to prove this- not general enough.

    What is B*(u+ 0v)? (u is a vector, 0v is the 0 vector)
    What is u+ 0v?

    If a is not 0, what is (aX)/a? If aX= 0 what does the answer to my question tell you?
     
  6. Sep 28, 2005 #5
    I'm really sorry but I still don't see how to show that
    B*0v = 0v
    --where B is a constant.

    Are you suggesting that I add the vector u to each side of the eqn?
     
  7. Sep 29, 2005 #6

    lurflurf

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    Homework Helper

    what are your axioms?
    One possible set includes
    (for a, b scalars and v a vector)
    (ab)v=a(bv)

    to show for some vector v
    v=0
    show that for any vector u
    u+v=v+u=u
     
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