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Linear Algebra: Linear dependence of vectors?

  1. Feb 17, 2008 #1
    1. The problem statement, all variables and given/known data
    Given:
    [tex]v_1 = \left(\begin{array}{cc}1\\-5\\-3\end{array}\right)

    v_2 = \left(\begin{array}{cc}-2\\10\\6\end{array}\right)

    v_3 = \left(\begin{array}{cc}2\\-9\\h\end{array}\right)

    [/tex]

    For what value of h is [tex]v_3[/tex] in Span{[tex]v_1, v_2[/tex]} and for what value of h is {[tex]v_1,v_2,v_3[/tex]} linearly dependent?
    2. Relevant equations



    3. The attempt at a solution
    If my understanding is correct, a vector is in a span of another set of vectors if all vectors are in the same plane? If so then my guess is h would have to be a multiple of -3.

    The linear dependence part of the question is getting me though. I was under the impression that all three vectors would have to be multiples of each other for it to be linearly dependent?
     
  2. jcsd
  3. Feb 17, 2008 #2
    For the first question.
    How can you express the vector space [itex]V=Span\{u_1,u_2\}[/itex]?
     
  4. Feb 17, 2008 #3
    A plane?
     
  5. Feb 17, 2008 #4
    Yes, and the equation that desribes that plane, is which?
     
  6. Feb 17, 2008 #5

    HallsofIvy

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    You understanding is not quite correct. A vector is in a span of another set of vectors if it can be written as a linear combination of the vectors in the set. The span of the set of vectors is a plane only if there are exactly two independent vectors in the set. That is not true in this problem!

    Are these really the problem you were given? Have you copied the vectors correctly? Notice that -2v1= v2. The span of these two vectors has dimension 1, not 2. v3 will be in their span only if it also is a multiple of either v1 or v2. The first component of v3, 2, is 2 times the first component of v1 (and -1 times the first component of v2) but the second component of v3 is -9 which is not 2 times the second component of v1 (and not -1 times the second component of v2).

    v3 while not be in the span of v1 and v2 for any value of h.


    (If the second component of v3 were -10, then v3 would be in the span of v1 and v2 for h= -6.)
     
  7. Feb 17, 2008 #6
    Ahhh, Ok. That makes a lot more sense.

    Yes, these really are the vectors.

    Just because making assumptions has burned me in the past, you mean the span is a line, correct?

    After going back and re-reading the chapter this is what I thought. Thanks for confirming it though.

    Since v3 isn't a linear combination of v1 and v2, this means that the set of {v1, v2, v3} has to be linearly independent, right?
     
  8. Feb 18, 2008 #7

    Defennder

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    Homework Helper

    Yes, the linear span of v1 and v2 is a line in R3.

    What does it mean for the set to be linearly independent? If it is linearly independent, you should not be able to represent any vector as a linear combination of the other 2. But haven't you just shown (or Halls did) that v1 and v2 are multiples of each other?
     
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