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Linear algebra: nonhomogeneous system

  1. Mar 28, 2007 #1
    1. The problem statement, all variables and given/known data
    Let A = [tex]\left( \begin{array}{l} \begin{array}{*{20}c} 0 & 1 & { - 1} \\ \end{array} \\ \begin{array}{*{20}c} 2 & 1 & 1 \\ \end{array} \\
    \end{array} \right)[/tex]. Suppose that for some b in [tex]\mathbb{R}^2[/tex], [tex]p = \left( {\begin{array}{*{20}c} 1 \\ { - 1} \\ 1 \\ \end{array} } \right)[/tex] is one particular solution to the nonhomogeneous equation Ax = b. What is the general form of the solution to this equation (Ax = b)?


    2. Relevant equations
    A relevant definition: A nonhomogeneous system is one with Ax = b, b is not equal to zero. A is a matrix and x and b are vectors.


    3. The attempt at a solution
    Since a homogeneous equation is Ax = 0, b = 0 in this particular question. Given p, I can presume that the general form of the solution will include p, but I'm not sure how to proceed.
     
  2. jcsd
  3. Mar 28, 2007 #2
    The solution is:

    [tex]\left( {\begin{array}{*{20}c}
    1 \\
    { - 1} \\
    1 \\

    \end{array} } \right) + x_3 \left( {\begin{array}{*{20}c}
    { - 1} \\
    1 \\
    1 \\

    \end{array} } \right),{\text{ }}x & _3 \in \mathbb{R}[/tex]

    But I can't make sense out of it.
     
  4. Mar 28, 2007 #3

    Hurkyl

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    Well, first off you can verify that actually is a family of solutions, right?

    You seem to think the homogeneous equation Ax=0 has some bearing; well, what is the solution space to Ax=0?
     
  5. Mar 28, 2007 #4
    I'm guessing since it's in the solution, the solution space is contained in R, but I'm not sure how to verify that it is actually a family of solutions. I'm trying to understand the presentation of the solution in terms of two vectors, one being multiplied by x3 for some reason.
     
  6. Mar 28, 2007 #5

    Hurkyl

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    Multiply it (on the left) by A. What do you get?
     
  7. Mar 29, 2007 #6

    daniel_i_l

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    Gold Member

    Given one solution and all the solutions to the respective homogeneous equation how do you find the solutions to the non-homogeneous equation?
    HINT: what happens if you add the two equations together?
     
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