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Homomorphism space/Linear subspaces/Fact

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Let and be vector spaces over a field . Then the following subsets are linear subspaces of .

  1. For a linear subspace ,

    is a linear subspace of . If and are finite-dimensional, then

  2. For a linear subspace ,

    is a linear subspace of , which is isomorphic to . If and are finite-dimensional, then

  3. For linear subspaces and ,

    is a linear subspace of . If and finite-dimensional, then

  4. For linear subspaces and ,

    is a linear subspace of .