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Math Help - Showing A Subset Is A Subspace

  1. #1
    Junior Member
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    Showing A Subset Is A Subspace

    Which of the following subsets of C^3 (complex) are subspaces:

    a) {(a,b,c) in C^3 | c=0}

    b) {(a,b,c) in C^3 | |a| is less than or equal to |b|}

    c) {(a,b,c) in C^3 | |a^2| + |b^2| + |c^2| =0}

    I'm having trouble understanding how to use vector addiction and scalar multiplication to see if they are subspaces or not. Any help is appreciated.
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  2. #2
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    Are you familiar with this criteria for vector subspaces:

    Let W\subseteq V be a subset of a vector space V over a field F. W is subspace of V if and only if:
    1. W\neq \emptyset.
    2. For all w_{1},w_{2}\in W, \lambda _{1},\lambda _{2}\in F exists \lambda _{1}w_{1} + \lambda _{2}w_{2}\in W.

    Try to use this for your problem.
    Here's an example for the first part:

    First you determine that the given set is not empty, well clearly \theta is of the form (a,b,0), so \theta \in A so A \neq \emptyset.

    Second, let w_{1}=(a,b,0), w_{2}=(d,e,0)\in A be some arbitrary vectors and let \lambda _{1},\labmda _{2} be some complex scalars. now \lambda _{1}w_{1} + \lambda _{2}w_{2} = (\lambda _{1}a+\lambda _{2}d,\lambda _{1}b+\lambda _{2}e,0)\in W.
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