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Linear Algebra: Linear Equation System -> Parameter Form

  1. Jul 8, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider the following linear system of equations:
    x1+2x3-5x4 = 0
    x1 + 4x2 +4x3 – 5x4 = 10
    x1 + 2x2 + 3x3 – 5x4 = 5
    4x1 + 2x2 + 9x3 – 20x4 = 5

    b) Solve the equation system with the Gaussian method.
    c) The solution set describes a plane. Specify it in the parameter form.


    2. Relevant equations
    Parameter form: [itex]\vec{r}[/itex]= [itex]\vec{r}[/itex]0 + λ[itex]\vec{v}[/itex] + β[itex]\vec{w}[/itex]


    3. The attempt at a solution
    3. My Work
    After Gauss?:
    1 2 0 -5 0
    0 2 4 0 10
    0 0 3 0 5
    0 0 0 0 0

    Rank is 3 so there is one free chooseable variable. I chose x4 and ended with the equation:
    [itex]\vec{X}[/itex] =
    [tex]\begin{vmatrix}-10/3\\5/3\\5/3\\0\end{vmatrix}[/tex] + [itex]\vec{x}[/itex]4 * [tex]\begin{vmatrix}5\\0\\0\\1\end{vmatrix}[/tex]

    Which I think is the plane’s formula? I’m not totally sure. My teacher said that he wouldn’t be putting the answers up, so I wanted a second opinion on my numbers. Please and thank you.
     
    Last edited: Jul 8, 2012
  2. jcsd
  3. Jul 8, 2012 #2

    chiro

    User Avatar
    Science Advisor

    Hey daemon_dkm and welcome to the forums.

    Plugging this into octave, I got the following reduced row echelon form:

    1.00000 0.00000 2.00000 -5.00000 0.00000
    0.00000 1.00000 0.50000 0.00000 2.50000
    0.00000 0.00000 0.00000 0.00000 0.00000
    0.00000 0.00000 0.00000 0.00000 0.00000

    This means setting x4 as our dummy we get x2 + x3 = 2.5 and x1 + x3 = 5x4 which means that adding the two together gives us x2 + x3 + x1 + x3 = 2.5 + 5x4 then minus x4 to LHS gives the plane equation:

    x1 + x2 + 2x3 - 5x4 - 2.5 = 0, which can be converted to n . (r - r0) = 0 where n is the normal vector, r is an arbitrary point and r0 is a vector one the plane. the normal is anything that is a multiple (except 0) of the vector (1,1,2,-5) and n.r0 = 2.5. where you can solve for any value of r0 you wish.

    Do you know how to go from ax + by + cz + dw + e to other formulas?
     
  4. Jul 8, 2012 #3
    ax + by + cz + dw + e

    the normal vector from that plane equation is n= (a, b, c, d), so then you need to find a vector that's perpendicular to the normal vector. Then you need to find another vector that's perpendicular to both of them to use them for the parameter form?
     
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