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Proving a set of matrices is NOT a vector space

  1. Apr 25, 2016 #1
    1. The problem statement, all variables and given/known data
    Show that the following is NOT a vector space:
    {(a, 1) | a, b, c, ∈ ℝ}
    {(b, c)

    Note: this is is meant to be a 2x2 matrix. This may not have been clear in how I formatted it.

    2. The attempt at a solution
    I am self-studying linear algebra, and have had a difficulty conceptualizing/interpreting what a vector space actually is. However, to show that the set of all matrices of the form above is NOT a vector space, I subjected it to the addition/multiplication conditions. For example, the condition v+w ∈ V:
    (a, 1) + (p, 1) = (a+p, 2)
    (b, c) + (q, r) = (b+q, c+r)

    The sum of the two matrices yields a matrix NOT of the original form, with 1 in the a_1,2 position. Does this prove that the set is not a vector space? Have I correctly interpreted the purpose of this problem as: "Show that one of the vector space conditions does not give back a matrix with the same form as the original matrix (a, 1, b, c)"?
    Another way I interpreted the question was: "Find a way to make a, b, or c NOT a real number." However, I could not think of a way to transform them into an imaginary number without using a scalar r ∉ ℝ.

    My book gives many examples of what vector spaces are, or can be, but leaves it up to the reader to infer what does not constitute a vector space.

    So, have I correctly solved the problem? Any helpful tips would be greatly appreciated! No answers, please. :)

    Thanks a lot,
    Jordan
     
  2. jcsd
  3. Apr 25, 2016 #2

    LCKurtz

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    Yes. That's all you needed. Even just the first equation was enough.
     
  4. Apr 25, 2016 #3

    Ray Vickson

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    Yes, your argument is correct. Also: a vector space must contain the zero vector, which your set of matrices does not.
     
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