Linear Algebra: Linear Equation System -> Parameter Form

[daemon_dkm]

Homework Statement

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.

Homework Equations

Parameter form: $\vec{r}$= $\vec{r}$₀ + λ$\vec{v}$ + β$\vec{w}$

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:
$\vec{X}$ =
$$\begin{vmatrix}-10/3\\5/3\\5/3\\0\end{vmatrix}$$ + $\vec{x}$₄ * $$\begin{vmatrix}5\\0\\0\\1\end{vmatrix}$$

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.

 

[chiro]

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?

 

[daemon_dkm]

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?