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Linear: Find a set of basic solutions and show as linear combination

  1. Jan 28, 2015 #1
    1. The problem statement, all variables and given/known data
    Find a set of basic solutions and express the general solution as a linear combination of these basic solutions

    a + 2b - c + 2d + e = 0
    a + 2b + 2c + e = 0
    2a + 4b - 2c + 3d + e = 0

    2. Relevant equations
    3. The attempt at a solution

    i reduced it to:
    1 2 0 0 -1 0
    0 0 1 0 2/3 0
    0 0 0 1 1 0

    a = -2s + t
    c = -2/3t
    d = -t

    I'm just not sure how i find solutions now. It could be literally anything could it not?
     
  2. jcsd
  3. Jan 28, 2015 #2

    LCKurtz

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    Assuming your arithmetic is correct (I didn't check), you have let the free variables ##b = s## and ##e = t##. I'm going to leave them as ##b## and ##e##. Write your solution as$$
    \left (\begin{array}{c}
    a\\b\\c\\d\\e
    \end{array}\right)
    =
    \left (\begin{array}{c}
    -2b+e\\b\\-\frac 2 3 e\\-e\\e
    \end{array}\right)
    =
    b\left (\begin{array}{c}
    ?\\?\\?\\?\\?
    \end{array}\right)
    + e
    \left (\begin{array}{c}
    ?\\?\\?\\?\\?
    \end{array}\right)
    $$Fill in the ?'s and you will have it.
     
    Last edited: Jan 29, 2015
  4. Jan 28, 2015 #3
    Do I just pick any number to plug in to b and e, and then pick different numbers and plug them into the c bracket?
     
  5. Jan 28, 2015 #4
    Do I just want to make it so each equation equals 0?
     
  6. Jan 29, 2015 #5

    LCKurtz

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    The ##c## in front of the last bracket was a typo; I have corrected it to ##e##. Fill in the ?'s and there is nothing left to do. Every solution can be gotten for some value of ##b## and ##e##. That is the general solution.
     
    Last edited: Jan 29, 2015
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