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Math Help - Subspaces--Linear Algebra

  1. #1
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    Subspaces--Linear Algebra

    I am trying to work on these proofs and I am having a hard time with this one.

    Let V be a vector space and S={v1, v2, v3,vk} be a set of vectors in V. Prove that span S is a subspace of V.

    I am not really sure where to even start. I know what span and I know that in order to prove something is a subspace of something else you must show u+v is an element of W and cu is an element of W. I just don't know what to do for this one.
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  2. #2
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    Quote Originally Posted by dude15129 View Post
    I am trying to work on these proofs and I am having a hard time with this one.

    Let V be a vector space and S={v1, v2, v3,vk} be a set of vectors in V. Prove that span S is a subspace of V.

    I am not really sure where to even start. I know what span and I know that in order to prove something is a subspace of something else you must show u+v is an element of W and cu is an element of W. I just don't know what to do for this one.
    You have said excatly what you need to do.

    First show it is non empty.

    \vec 0 \in span(S) becuase
    \vec 0= 0v_1+0v_2+... + 0v_k

    done

    Now we need to show it is closed. So let \vec a, \vec b \in span(S)

    Then a=a_1v_1+...a_nv_n and
    b=b_1v_1+...+b_nv_n
    Then
    a+b=(a_1v_1+...a_nv_n)+(b_1v_1+...+b_nv_n)=(a_1+b_  1)v_1+...+(a_n+b_n)v_n \in span(S)

    I think you can show the last one. Good luck
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