Question

Find the bases for the following subspace of R^3:
The plane 3x - 2y + 5z = 0

Answer 1

The plane is a two dimensional subspace in R^3, so you need to find any two linearly independent vectors that lie in the plane.

Let's find one which has no z component. Setting x = 1, then
3 - 2y + 0 = 0
=> (x,y,z) = (1,3/2,0) is one vector in the plane.

Now just find another and provided it is not a scalar multiple of the vector we just found, the two vectors together will form a basis.

That basis won't be orthonormal necessarily, but that isn't what the question asked; it's just asking for any basis.