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Math Help - Matrix Representation of Linear Transformations (2)

  1. #1
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    Matrix Representation of Linear Transformations (2)

    Let V and W be vector spaces, and let S be a subset of V. Define S^0 = {T Є L (V,W): T(x) = 0 for all x Є S}. Prove the following statements:

    a) S^0 is a subspace of L(V,W)
    b) If S1 and S2 are subsets of V and S1 is contained in S2, then S2^0 is contained in S1^0


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  2. #2
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    Let V and W be vector spaces, and let S be a subset of V. Define S^0 = {T Є L (V,W): T(x) = 0 for all x Є S}. Prove the following statements:

    a) S^0 is a subspace of L(V,W)
    To show S^0 is a subspace you need to prove T_1,T_2 \in S^0 then T_1 + T_2\in S^0. Next you need to prove that if T \in S^0 and k \in K (the field you are working over) then kT \in S^0

    b) If S1 and S2 are subsets of V and S1 is contained in S2, then S2^0 is contained in S1^0
    To show S_2^0 \subseteq S_1^0 you need to show if T \in S_2^0 then T \in S_1^0.
    This is straightforward.
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