Basis of a subspace defined by a linear equation. Find a basis of the subspace of $${\mathbb R}^3$$ defined by the equation $$-9 x_1 + 3 x_2 + 2 x_3 = 0$$
I'm looking on how to approach this problem since my instructor only showed us how to prove if they are linearly independent or not and I can't find any sources on line.. Thanks for the assist.

You were given one equation and three variables. It would mean that you get two free parameters (your final space will have dimension=2). You can choose for example $x_1=1,x_2=0$ getting $x_3=9/2$ and $x_1=0,x_2=1$ obtaining $x_3=-3/2$ (this choice is generally free). Then, two independent vectors of the subspace are:
$$\left(1,0,\frac{9}{2}\right)\\
\left(0,1,\frac{-3}{2}\right)$$
Finally, you can take care about aesthetics and eliminate the fractions multiplying by two
$$\left(2,0,9\right)\\
\left(0,2,-3\right)$$
Any other vector you can find in your subspace will be linearly dependent on these two the vectors
