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Linear algebra question.

  1. Feb 9, 2013 #1
    1. The problem statement, all variables and given/known data

    We are to show that the set C of complex numbers, with scalar multiplication de ned
    by α (a + bi) = α a + α bi and addition de fined by (a + bi) + (c + di) = (a + c) + (b + d)i,
    satis es the eight axioms of a vector space

    I have a few questions about this problem,

    What is the term i? is it just a fancy way of saying a2

    Can we think of these as vectors, for instance (a+bi)
    is the vector X where a is x1 and bi is x2?

    Also I was trying to prove the third axiom which states there exist an element 0 in V such that x+0=x for each xεV.

    My logic was let (a+bi)= vector X and (c+di)= Vector Y
    X+Y=X

    X-X+Y=X-X
    Y=0
    thus X+Y=X

    Thanks for the help guys.
    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 9, 2013 #2

    Dick

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    i is the imaginary unit. i^2=(-1). But all you really need to know about complex numbers is that a+bi=c+di if and only if a=c and b=d. And, sure, the additive identity is 0 or 0+0i. Same thing.
     
  4. Feb 9, 2013 #3
    Thanks for the response Dick. So my logic and reasoning was perfectly find for this problem?
     
  5. Feb 9, 2013 #4

    Dick

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    You deduced the identity is 0+0i, sure. Showing it is an identity is just a matter of saying (0+0i)+(a+bi)=(0+a)+(0+b)i=a+bi. That's only one axiom though. Haven't you got seven more to go?
     
  6. Feb 9, 2013 #5
    Yea, but I just wanted to make sure I was on the right track. Thanks!
     
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