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Linear Combinations

  1. Sep 12, 2010 #1
    Will there always be two different combinations that produce b=(0,1) of three vectors: u, v, and w?

    I'm pretty certain that the answer is no, but am I right in saying that with three vectors, assuming they are not all parallel, will always have at least one combination that produces (0,1)
     
  2. jcsd
  3. Sep 12, 2010 #2

    Dick

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    You've got that right. If you have three vectors in R^2 then either there are no combinations that produce (0,1) (if they are all parallel and not parallel to (0,1)) or there are an infinite number.
     
  4. Sep 12, 2010 #3
    How can there be an infinite number?
     
  5. Sep 12, 2010 #4

    Dick

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    Say u=(0,1), v=(1,0) and w=(1,1). There are an infinite number of solutions to the equation a*u+b*v+c*w=(0,1). You should be able to show that. It's maybe a little harder to show that if there is one solution, then there are an infinite number.
     
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