what is the basis of a subspace or R3 defined by the equation

Question

2x1+3x2+x3=0

I know the basis is vectors <-3,2,0> and <-1,0,2>
but how do you get that?

Answer 1

The equation:
2x1+3x2+x3=0
Defines a plane.
Any two different (not linearly dependent) vectors in that plane form a basis.
Pick any old values for x and y then solve for z
like 1,1 then -5
and 1,-1 then 1
so I would say
(1,1,-5) and (1,-1,1)
the ones you gave are
-3,2 then you get -6+6+x3=0 so x3= 0 sure enough (-3,2,0)
If you chose -1,0 then you would get x3 = 2 so (-1,0,2)
both answers and an infinite number of other answers, are fine.