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Vector Spaces

  1. Apr 20, 2010 #1
    1. The problem statement, all variables and given/known data
    Determine if the following sets are vector spaces

    Part A) R^2; u+v = (u1 + v1, u2 + v2); cu = (cu1,cu2)

    part b) M_2,2

    A + B = |(a_11 + b_11) (a_12 + b_12)| & cA = |ca_11 ca_12|
    |(a_21 + b_21) (a_22 + b_22)| |ca_21 ca_22|

    sorry this is a little crude but those are both suppose to be 2X2 matricies


    2. Relevant equations



    3. The attempt at a solution
    Part A) Yes, it is a vector space because it follows the addition & multiplication properties of vector spaces

    Part B) yes it is a vector space because the set is closed under matrix addition & sclar multiplication


    Are these statments true and a sufficient answer
     
  2. jcsd
  3. Apr 20, 2010 #2
    If they both are closed under scalar multiplication and addition, then you can conclude that they might be a subset of a given vector space. To check whether or not they are vector spaces, you have to go through the 10 axioms. Does it sound familiar?
     
  4. Apr 20, 2010 #3
    Your solution is not sufficient because you need to go through each and every vector space axiom to prove that it is in fact a vector space (and you need to show where only one axiom fails to prove it is not a vector space).

    It is easy to see that pointwise addition is a simple procedure, but I think the point of the exercise is to go through the seven vector space axioms - many are trivial but as far as it stands now your answer is not sufficient.
     
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