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Linear Algebra-Linear Spaces

  1. Nov 3, 2009 #1
    1. The problem statement, all variables and given/known data
    Are the 3X3 matrices A such that vector <1,2,3> is in the kernel of A, a subspace of R^(3X3)?

    2. Relevant equations

    3. The attempt at a solution
    I know that the kernel condition gives a subset V={A|A*<1,2,3>=0} but im not sure of how to proceed to show it is in fact a subspace. Should I try to see if it spans or use the definition of a subspace? i'm just confused on how to proceed with the next step
  2. jcsd
  3. Nov 3, 2009 #2
    To show that it's a subspace, you have to show that if it contains u and v, then it contains au + bv where a and b are scalars, that it contains 0 and that it contains -u for any u that it contains.
  4. Nov 3, 2009 #3


    Staff: Mentor

    To expand slightly on what aPhilosopher said, let U and W be matrices in your subset V. It's sufficient to show that aU + bW is also in the same subset, for any scalars a and b.

    Showing that the subset contains the zero matrix corresponds to a = b = 0. Showing that -U is in the subset corresponds to a = -1 and b = 0.
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