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For what values of a are these vectors linearly dependent?

  1. Apr 10, 2013 #1
    1. The problem statement, all variables and given/known data
    For what value(s) of a are the following vectors v1=[1,2,-1], v2=[0,1,3], and v3=[a,4,5] linearly dependent?

    2. Relevant equations

    Since linear dependence means that any one of the vectors can be expressed as a linear combination of the other vectors:
    sv1v1+s2v2=v3

    where s1and s2 are some coefficients.

    3. The attempt at a solution

    Looking at the vectors, I tried v1+2v2=v3

    Solving, I get a=1

    I'm fairly certain I did the question correctly. The only thing that really bothers me is the plural of "value" in the question - is there ever a case where there can be more than one solution? How can I tell and solve those cases?
     
  2. jcsd
  3. Apr 10, 2013 #2
    The vectors [a,4,5] define a plane. So do [1,2,-1] and [0,1,3]. The linearly dependent cases are the intersections of these two planes.
     
  4. Apr 10, 2013 #3

    ehild

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    It is the only solution, but instead of trial and error, you can find the solution(s) by solving the equation valid for the components of the vectors.

    ehild
     
  5. Apr 10, 2013 #4
    Doesn't that imply that any one of the vectors can be expressed as a linear combination of the other vectors? Basically equate the two planes and solve (which I did)?

    Could you elaborate, please? What equation?

    I did Gaussian Elimination and wound up with a free variable. How am I supposed to solve for a specific case other than to plug in random values (which is essentially what I did in the first place with far less work)? How does it help me determine all of the possible solutions in other cases?
     
  6. Apr 10, 2013 #5

    ehild

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    s1v1+s2v2=v3. That means three equations for the x,y,z components

    s1=a
    2s1+s2=4
    -s1+3s2=5.

    Three equations with three unknowns. No free parameters.

    ehild
     
  7. Apr 10, 2013 #6
    Yes. Two planes intersect either on only one line, or everywhere.
     
  8. Apr 10, 2013 #7
    Ah. Just solve the matrix/system of equations for a.

    Thank you.
     
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