Understanding Vector Spaces

Discussion in 'General Math' started by Ken, Aug 15, 2007.

  1. Ken

    Ken Guest

    I answered the following two questions as true...
    a) in any vector space ax=bx implies that a = b. Where a,b are real
    numbers and x is a vector.
    b) In any vector space ax = ay implies that x = y. Where a is a real
    number and x and y are vectors.
    How can these be false?
    and there is one more I don't follow...
    c) If f and g are polynomials of degree n, then f + g is a polynomial
    of degree n. (I also said this was true but it seems the answer is
    false).

    If you wish to reference the conditions of vector spaces... [1-8] or
    simply state that a is in violation of any of these conditions it is
    here for reference: http://en.wikipedia.org/wiki/Vector_spaces
     
    Ken, Aug 15, 2007
    #1
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  2. For a) let x be the zero vector. For b) let a be the zero scalar.

    For part c) suppose f(x) = x, and g(x) = -x, what is the degree of
    f+g?

    Brian
     
    Brian VanPelt, Aug 16, 2007
    #2
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  3. Ken

    Ken Guest


    Thank you for the help... I will from now on all ways consider the
    zero case.
    The degree is -1.
     
    Ken, Aug 30, 2007
    #3
  4. Wow, it's been a long time since I have written this message and I
    forgot about it.

    This brings up an interesting point though - the degree of the zero
    polynomial. Did you know that some people don't bother to define the
    degree of the zero polynomial? But yet, some people give it degree
    -1, as you do? Could it be -2, -pi, or otherwise? Could the degree
    of the zero polynomial be i (ooh, i has magnitude equal to 1)? Every
    stance has its reasons, but I have never seen one stance that
    logically crushes all others, and it's quite possible that there is no
    logically correct stance here.

    Thanks,

    Brian
     
    Brian VanPelt, Aug 31, 2007
    #4
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