In linear algebra, a basis for any vector space is formally defined as a minimal generating set that satisfies both spanning and linear independence conditions. For homogeneous systems ($Ax = 0$), the solution vectors derived from free variables via Gaussian elimination constitute a guaranteed basis for the nullspace due to their inherent structure. Conversely, in non-homogeneous contexts or when analyzing column spaces directly on row-reduced echelon forms (RREF), the columns containing leading ones form a minimal spanning set that is inherently linearly independent, thereby constituting a basis for the colunnspace of the transformation matrix.
In linear algebra, a basis for any vector space is formally defined as a minimal generating set that satisfies both spanning and linear independence conditions. For homogeneous systems ($Ax = 0$), th…