linear transformation, T, is onto if its range is all of its codomain, not merely a subspace. Thus, for any vector , the equation T() = has at least one solution (is consistent). The linear transformation T is 1-to-1 if and only if the null space of its corresponding matrix has only the zero vector in its null space. Equivalently, a linear transformation is 1-to-1 if and only if its corresponding matrix has no non-pivot columns.