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Math Help - Linear combination is also in the span

  1. #1
    Senior Member sfspitfire23's Avatar
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    Linear combination is also in the span

    Suppose that vectors x and y are both in the linear span of the
    vectors r1, r2, r3. Show that any linear combination of x and y is also
    in the linear span of r1, r2, r3.

    What I have-

    Because x and y are in the span of r1,r2,r3, then x and y are linear combinations of r1,r2,r3. So, the following exists:

    c1(r1)+c2(r2)+c3(r3) = x

    and

    d1(r1)+d2(r2)+d3(r3) = y

    where the d's and c's are scalars. I'm not sure this gets me anywhere with the proof.

    Thanks.
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  2. #2
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    Re: Linear combination is also in the span

    if:

    x = c_1r_1 + c_2r_2 + c_3r_3, and y = d_1r_1 + d_2r_2 + d_3r_3,

    then ax+by = a(c_1r_1 + c_2r_2 + c_3r_3) + b(d_1r_1 + d_2r_2 + d_3r_3)

    = (ac_1)r_1 + (ac_2)r_2 + (ac_3)r_3 + (bd_1)r_1 + (bd_2)r_2 + (bd_3)r_3

     = (ac_1+bd_1)r_1 + (ac_2+bd_2)r_2 + (ac_3+bd_3)r_3 which is surely in span({r1,r2,r3})

    since for each i, ac_i + bd_i is surely some scalar.
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  3. #3
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    Re: Linear combination is also in the span

    Quote Originally Posted by sfspitfire23 View Post
    Suppose that vectors x and y are both in the linear span of the vectors r1, r2, r3. Show that any linear combination of x and y is also
    in the linear span of r1, r2, r3.
    Suppose that x = \chi _1 r_1  + \chi _2 r_2  + \chi _3 r_3 \;\& \,y = \gamma _1 r_1  + \gamma _2 r_2  + \gamma _3 r_3 .

    Show that \alpha x + \beta \,y = \omega _1 r_1  + \omega _2 r_2  + \omega _3 r_3 .

    i.e. \omega _k=~?
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