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Linear Independence

  1. Sep 16, 2010 #1
    How can I prove given an arbitrary set of vectors v1 and v2, given they are linearly independent, that their sum (v1 + v2) is also linearly independent?
     
  2. jcsd
  3. Sep 16, 2010 #2

    Office_Shredder

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    Linearly independent from what?
     
  4. Sep 16, 2010 #3
    let me rephrase that, how can i show that v1,v2, and v1+v2 is linearly independent given v1 and v2 is linearly independent. I seemed to have left out a key statement.
     
  5. Sep 16, 2010 #4

    Office_Shredder

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    You can't. Try to find the linear dependency
     
  6. Sep 16, 2010 #5
    I see. How can I prove otherwise?
     
  7. Sep 17, 2010 #6

    Office_Shredder

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    Well, how do you prove that a set of vectors is linearly dependent?
     
  8. Sep 17, 2010 #7
    [tex]\vec{v_{1}}, \vec{v_{1}}, \mbox{ and }\vec{v_1}+\vec{v_2}[/tex] are linearly independent if the only solution to

    [tex]a\vec{v_{1}}+b\vec{v_{2}}+c(\vec{v_{1}}+\vec{v_{2}})=0[/tex]

    is [tex]a=0, b=0, c=0[/tex].

    Is this the case, or can you find other values that satisfy this equation?
     
  9. Sep 17, 2010 #8

    Mark44

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    The set {v1, v1, v1 + v2 } is always linearly dependent, since the third one listed is a linear combination of the first two.
     
  10. Sep 17, 2010 #9

    HallsofIvy

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    Well, that reduces to (a+ c)v1+ (b+ c)v1= 0. Since v1 and v2 are independent, you must have a+ c= 0 and b+ c= 0. Obviously a= b= c is one solution to that but those are only two equations in three unknows. We can typically solve two equations in two unknowns. Okay, solve for a and b, say, treating c as a number. Then let c be whatever you want.

    Solving for a and b "in terms of c" gives a= -c and b= -c. Take c to be anything you like and find a and b. If you happen to select c= 0, then, of course, you get a= b= c= 0. But what if you select c= 1?
     
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