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Math Help - Plz.. help with linear algebra

  1. #1
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    Plz.. help with linear algebra

    let u and v be distinct vectors of a vector space V. Show that if {u,v} is a basis for V and a and b are nonzero scalrs, then both {u+v,au} and {au,bv} are also bases for V.

    Please help, i dont even know where to start!!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by mathlovet View Post
    let u and v be distinct vectors of a vector space V. Show that if {u,v} is a basis for V and a and b are nonzero scalrs, then both {u+v,au} and {au,bv} are also bases for V.

    Please help, i dont even know where to start!!
    well, i'll do the first one for you and you do the second one..

    i) suppose u+v and au are linearly dependent. then there exists a nonzero scalar k such that u+v = kau

    then, u+v - kau = (1-ka)u + v = 0

    this means that v = (ka-1)u which contradicts the fact that { u,v} is a basis for V.

    hence, u+v and au must be linearly independent.

    ii) Let w\in V. since { u,v} is a basis for V, we have w = mu + nv for some scalars m and n in V. In particular, let m=M+Na and n=M where M and N are scalars in V. Hence, w = mu + nv = (M+Na)u + Mv = Mu + Nau + Mv = M(u+v) + Nau. Thus, w\in \mbox{span}(\{u+v, au\}) implies V\subseteq \mbox{span}(\{u+v, au\}) since we have written w as a linear combination of \{u+v, au\}. Clearly, \mbox{span}(\{u+v, au\}) \subseteq V. Therefore, \mbox{span}(\{u+v, au\}) = V.
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