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Math Help - linearly dependent

  1. #1
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    linearly dependent

    Hi guys, I need help with the following question:

    Given that {X1, X2,..., Xk} is a linearly independent set, show that {X1, X2,..., Xk, y} is linearly dependent if and only if y is in the span {X1, X2,..., Xk}.
    Last edited by amitmeh; April 19th 2009 at 06:37 PM.
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  2. #2
    Super Member Gamma's Avatar
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    Linearly Independent

    A set {x_1, x_2, ..., x_n} is said to be linearly independent by definition if a_1 x_1 + a_2 x_2 + ... + a_n x_n=0 implies a_i=0 for all i.

    Suppose {x_1, x_2,... x_n, y} are linearly dependent, then there exist coefficients, not all of which are 0, such that a_1 x_1 + a_2 x_2 + ... + a_n x_n + a_{n+1} y=0 . Notice that if a_{n+1} = 0 then we would contradict the linear independence of {x_1, x_2, ..., x_n}. Thus we can say y=-(a_2 x_2 + ... + a_n x_n)/a_{n+1} Showing it to be in the span as desired.

    Conversely if y is in the span of {x_1, x_2, ..., x_n} then there exist coefficients such that y=a_1 x_1 + a_2 x_2 + ... + a_n x_n but then we would have 0=a_1 x_1 + a_2 x_2 + ... + a_n x_n + (-1)y all of which are not zero, so {x_1, x_2,... x_n, y} is linearly dependent.

    QED
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  3. #3
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    Quote Originally Posted by Gamma View Post
    all of which are not zero...
    You mean "some of which..."?
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  4. #4
    Super Member Gamma's Avatar
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    Yeah

    yeah, i guess I can see the confusion there.
    I mean all of them cannot be zero, so yeah some of them are nonzero, same thing.
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  5. #5
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    thanks

    thank you so much!
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