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2 simple vector spaces question

  1. Apr 24, 2005 #1
    hi,i got 2 question about vector spaces :

    1. Do the set of all n-tuples of real numbers of the form (x, x1 ,x2.....xn) with the standard operation on R^2 are vector spaces?

    2.Do the set of all positive real numbers with operations

    x+y =x*y and kx=x^2 are vector space?
     
  2. jcsd
  3. Apr 24, 2005 #2

    matt grime

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    1 makes no sense. What do "the standard operations of R^2", which are undefined, have to do with the n-tuples of real numbers? R^n is a vector space.

    2, just try and verify the axioms, or figure out where they may go wrong.
     
  4. Apr 24, 2005 #3
    those question is from a book ...........
     
  5. Apr 24, 2005 #4

    HallsofIvy

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    And does the book use phrases like "those question"?

    Once again: your first question makes no sense. It would make sense if you asked "Does the set of all n-tuples of real numbers of the form (x1 ,x2.....xn) with the standard operation on R^n form a vector space?". In that case the anser is obviously "yes". It makes no sense to talk about " real numbers of the form (x1 ,x2.....xn) with the standard operation on R^2" because you can't apply the operations on R^2 to R^n.

    The second question, "2.Is the set of all positive real numbers with operations
    x+y =x*y and kx=x^2 a vector space?" is reasonable. Matt Grimes' point was that it is just a matter of checking the axioms for (or definition of) a vector space.
     
  6. Apr 25, 2005 #5
    so you are saying that :
    the set of all positive real numbers with operations
    x+y =x*y and kx=x^2 are vector space?
     
  7. Apr 25, 2005 #6

    HallsofIvy

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    No, we are saying you should check to see if the "axioms" for a vector space are satisfied yourself. In particular, is the "distributive law", k(x+y)= kx+ ky, satisfied?
     
  8. Apr 26, 2005 #7
    ( a 1 ) --- is this 2x2 matrix a vector space ? sorry for asking this coz this is abstract
    1 b me
     
  9. Apr 27, 2005 #8

    matt grime

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    Why is it abstract? What are the operations, for instance, what is k*M for some scalar k? Is that in the set? Check the rules, there really is nothing complicated or hidden in this:

    to check if S is a vector space over R, say, check the rules: is ks in S when s is in S and k is in R? ie if s satisfies the rules to be in S, does ks? Similarly what about s+t when s and t are in S? is there a zero vector? Just three simple rules to check.
     
  10. Apr 27, 2005 #9
    there are a few more rules to check too (like the afformentioned distributive property).
     
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