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Vector span

  1. Aug 22, 2012 #1
    1. The problem statement, all variables and given/known data

    is vector b in the span of vectors v1,v2? Give reasons.



    2. Relevant equations



    3. The attempt at a solution

    v1=(1,4,0,-1)

    v2=(2,7,-2,-3)

    b= (-1,-1,7,4)

    set up in matrix

    (v1,v2| b)

    and after row reduction I have

    (1,0,0,0),(2,-1,0,0)|(-1,3,6,6)

    matrix has no sltn

    since 0x + 0y = 6

    therefore b does not span v1 ,v2.

    my row reduction steps are..

    R4= R4+R1
    R2=R2-4R1


    R4=R4+R2
    R3=R3+R1


    .....


    is this right?
     
  2. jcsd
  3. Aug 22, 2012 #2
    The general result (negative) is correct. However, I get a different result numerically. Specifically, R3 = R3 + R1 seems wrong because that gives you 1 n the first column of R3, which you do not want.
     
  4. Aug 22, 2012 #3
    ohh thanks I did not see that.
     
  5. Aug 22, 2012 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    A more fundamental resolution is that a vector, b, is in the span of vectors v1 and v2 if and only if there exist scalars, a and b, such that b= av1+ bv2 (that's the definition of 'span').

    That is, there must exist a and b such that a(1, 4, 0, -1)+ b(2, 7, -2, -3)= (a+ 2b, 4a+ 7b, -2b, -a- 3b)= (-1, -1, 7, 4) so that we must have
    a+ 2b= -1
    4a+7b= -1
    -2b= 7
    -a- 3b= 4.

    Of course, what you are doing is using the 'augented' matrix to try to solve those equations. But in this simple case, you can recognize that the third equation, -2b= 7 gives b= -7/2 and so the fourth equation becoms -a+ 21/2= 4 so that a= 21/2- 4= (21- 8)/2= 13/2. Now put those values of x and y into the first and second equations to see if they satisfy them: a+ 2b= -1 becomes 13/2+ (-14/2)= -1/2, NOT -1 so we can stop.
     
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