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Math Help - Another Spanning set question

  1. #1
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    Another Spanning set question

    Given

    x1=[-1 2 3]^T x2=[3 4 2]^T x=[2 6 6]^T y=[-9 -2 5]^T

    Is x Є Span(x1,x2)?

    I'm not sure where to go with this. It doesn't seem as though x1 and x2 would span R^3 but I don't know if that would be relevant or not. Any help would be appreciated, thank you!
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  2. #2
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    Quote Originally Posted by pakman View Post
    Given

    x1=[-1 2 3]^T x2=[3 4 2]^T x=[2 6 6]^T y=[-9 -2 5]^T

    Is x Є Span(x1,x2)?

    I'm not sure where to go with this. It doesn't seem as though x1 and x2 would span R^3 but I don't know if that would be relevant or not. Any help would be appreciated, thank you!
    What we need to know is does there exits a,b \in \mathbb{R}

    ax_1+bx_2=x \iff (-a,-2a+3a)+(3b,4b,2b)=(2,6,6)

    or the system

    -a+3b=2
    -2a+4b=6
    3a+2b=6

    The above overdetermined system is inconsistant

    So x is not in the span.

    I hope this helps
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  3. #3
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    Quote Originally Posted by pakman View Post
    Given

    x1=[-1 2 3]^T x2=[3 4 2]^T x=[2 6 6]^T y=[-9 -2 5]^T

    Is x Є Span(x1,x2)?

    I'm not sure where to go with this. It doesn't seem as though x1 and x2 would span R^3 but I don't know if that would be relevant or not. Any help would be appreciated, thank you!
    Essentially you should be asking:

    Does there exist reals a and b such that ax1 + bx2 = x?
    That translates to these equations:
    -a+3b = 2 ---------(I)
    2a+4b = 6---------(II)
    3a+2b = 6---------(III)

    These are not consistent. This means x \not \in \text{ Span}(x_1, x_2)
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