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Math Help - Linear independence and span

  1. #1
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    Question Linear independence and span

    Q: If S is a linearly independent subset of a vector space V, then is it true in general that for no proper subset T of S, span(T) = span(S) ?

    The result should be true if S is a basis. But in the above question one can't make out whether or not S is a basis, right ? So what should be the answer ?

    Please help
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  2. #2
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    Re: Linear independence and span

    Quote Originally Posted by mrmaaza123 View Post
    Q: If S is a linearly independent subset of a vector space V, then is it true in general that for no proper subset T of S, span(T) = span(S) ?

    The result should be true if S is a basis. But in the above question one can't make out whether or not S is a basis, right ? So what should be the answer ?

    Please help
    Well, you can think of S as a basis for the subspace of V generated by S itself, so...
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  3. #3
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    Re: Linear independence and span

    Let S be the set of vectors \{v_1, v_2, ..., v_n\} and T the subset \{v_1, v_2, ..., v_m\} with m< n, of course. Obviously, v_n is in Span(S) so if Span(S)= Span(T) then we must have v_n= a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_mv_m. From that it follows that a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_mv_m- v_m= 0 contradicting the fact that the vectors in S were independent.
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