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Math Help - Subspace

  1. #1
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    Subspace

    I need help proving the following:

    Let F be any field. Let A be any m x n over F. Suppose that W is a subspace of F^n. Show that the set {Aw:w in W) is a subspace of F^m.

    Note that the lowercase w in the above set is the w vector.

    Thank you for any help.
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  2. #2
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    Quote Originally Posted by thesummerofgeorge View Post
    I need help proving the following:

    Let F be any field. Let A be any m x n over F. Suppose that W is a subspace of F^n. Show that the set {Aw:w in W) is a subspace of F^m.

    Note that the lowercase w in the above set is the w vector.

    Thank you for any help.
    Let S=\{ A \omega | \omega \in W \}
    Again the idea is the same as I said.
    We need to show closure.
    kA \omega =A (k\omega)=A \omega_1 \in S
    Where \omega_1=k\omega \in W
    (Because \omega is an element of a vector space over F thus, k\omega \in W.)

    A \omega_1+A\omega_2=A(\omega_1+\omega_2)=A \omega_3 \in S
    Where, \omega_3 \in W.
    Thus, the set S which is a subset of W is closed under scalar multiplication und vector addition, that is it must be a vector space (or subspace of W) over the field F.
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