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Homework Help: LA: Finding a basis

  1. Jul 7, 2011 #1
    1. The problem statement, all variables and given/known data

    Let S = [x y z w] [itex]\in[/itex] [tex]R^4[/tex] , 2x-y+2z+w=0 and 3x-z-w=0

    Find a basis for S.


    2. Relevant equations


    3. The attempt at a solution

    I started by putting the system into reduced row form:

    [2 -1 2 1]
    [3 0 -1 -1]

    [2 -1 2 1]
    [0 3 -8 -5]

    [6 0 -2 -2]
    [0 3 -8 -5]

    [1 0 -1/3 -1/3]
    [0 1 -8/3 -5/3]

    Now have:

    x - 1/3z - 1/3w = 0
    y - 8/3z - 5/3w = 0

    Letting z = s, and w = t, we get:

    x = 1/3s + 1/3t
    y = 8/3s + 5/3t
    z = s
    w = t

    And this gives:

    s[1/3 8/3 1 0] and t[1/3 5/3 0 1]

    Where the basis vectors are:

    [1/3 8/3 1 0] and [1/3 5/3 0 1]

    and are linearly independent.


    Did I do this correctly? I'm really struggling with these concepts and I feel like I'm missing something. Thanks in advance.
     
  2. jcsd
  3. Jul 7, 2011 #2

    LCKurtz

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    I didn't check all your arithmetic, but assuming your numbers are OK, yes. That is exactly how to work the problem.
     
  4. Jul 8, 2011 #3

    HallsofIvy

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    Personally, I would prefer this:
    2x-y+2z+w=0 and 3x-z-w=0

    From the second equation, w= z- 3x
    Putting that into the first equation, 2x- y+ 2z+ z- 3x= -x- y+ 3z= 0 so y= 3z- x.
    Having solved for y and w in terms of x and z,
    [x, y, z, w]= [x, 3z- x, z, z- 3x]= [x, -x. 0, -3x]+ [0, 3z, z, z]= x[1, -1, 0, -3]+ z[0, 3, 1, 1]
    so a basis is {[1, -1, 0, -3], [0, 3, 1, 1]} which is essentially what you have, multiplied by 3.
     
  5. Jul 8, 2011 #4
    Thanks for the replies. Although, I'm not sure I understand your basis HallfofIvy. I can't seem to find how it's related to the one I ended up with. I understand how you got there with the substitution, but is it just another basis for S?
     
  6. Jul 9, 2011 #5

    HallsofIvy

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    It is just your method using basic algebra to solve the equations rather than "row-reduction".

    Every member of this subspace is of the form <x, y, z, w> and we must have
    2x-y+2z+w=0 and 3x-z-w=0

    I just solved for y and w in terms of x and z and then used x and z as multipliers where you solved for x and y and used z and w as multipliers.
     
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