Basis for a Plane that is given.

1. The problem statement, all variables and given/known data

Consider the plane 3x₁-x₂+2x₃ = 0 in R³. Find a basis for this plane. Hint: It's not hard to find vectors in this plane.



2. Relevant equations

Plane: 3x₁-x₂+2x₃ = 0 in R³.



3. The attempt at a solution

Let,

A = [itex]\left[3,\right.\left.-1,\right.\left.2\right][/itex] [itex]\rightarrow[/itex] [itex]\left[\frac{3}{3},\right.\left.\frac{-1}{3},\right.\left.\frac{2}{3}\right][/itex] (Row Reduced)



[itex]\Rightarrow[/itex] x₁ = [itex]\frac{1}{3}[/itex]x₂-[itex]\frac{2}{3}[/itex]x₃, x₂ is free, x₃ is free.



[itex]\Rightarrow[/itex] [itex]\left\{x_2[\frac{1}{3},1,0]+x_3[\frac{-2}{3},0,1] | x_2, x_3 \in R\right\}[/itex]



[itex]\Rightarrow[/itex] Basis of Plane = [itex]\left\{[\frac{1}{3},1,0],[\frac{-2}{3},0,1]\right\}[/itex]