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Math Help - linear combination.. help please

  1. #1
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    Question linear combination.. help please

    Is vector u = [3 3 2] is a linear combination of vector v1 =[1 0 1] and
    vector v2 =[1 1 1] ?



    could you please show me how to solve this problem? thank you very much.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello,
    Quote Originally Posted by happystudent View Post
    Is vector u = [3 3 2] is a linear combination of vector v1 =[1 0 1] and
    vector v2 =[1 1 1] ?

    could you please show me how to solve this problem? thank you very much.
    This question can be answered using the determinant the three vectors : u is a linear combination of v_1 and v_2 iff the determinant D=\left|\begin{smallmatrix} 3&1&1\\3&0&1\\2&1&1\end{smallmatrix}\right| equals 0. Do we have D=0 ?

    Another way of answering this question is solving u=av_1+bv_2 for a,b\in\mathbb{R}. If this equation has at least one solution then u is a linear combination of v_1 and v_2. If it has no solution then...
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  3. #3
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    In order for vector u = [3 3 2] to be a linear combination of vector v1 =[1 0 1] and vector v2 =[1 1 1], it would necessary u=3[1,1,1]+t[1,0,1]. That is the only way to have 3 in the second position of u. But that means that 3+t=3 and 3+t=2. But that is impossible.
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