How to find a subspace's basis?
Given what information? How is the subspace defined? A basis for any vector space depends crucially on what the subspace is and how it is defined!
An example: find a basis for the subspace of all (x, y, z) in satisfying 3x- 2y+ 2z= 0.
We can solve that equation for z: 2z= 2y- 3x so z= y- (3/2)x. Any vector in that subspace can be written (x, y, z)= (x, y, y- (3/2)x)= (x, 0, -(3/2)x)+ (0, y, y)= x(1, 0, -3/2)+ y(0, 1, 1) so a basis for that subspace is {(1, 0, -3/2), (0, 1, 1)}.
It is important, of course, to understand what a "subspace" is! Do you understand why neither of the subsets "all (x, y, z) satisfying [tex]x^2+ y^2- z= 0" or "all x, y, z, satisfying 3x- 2y+ 2z= 1" is a subspace of ?
It is also important to understand that the is no one correct basis. Any vector space has an infinite number of possible bases.
If that does not help, please given an example you are having difficulty with.