The core principle establishes a condition where vectors in a vector space cannot be expressed as linear combinations of one another, characterized formally by the existence of only trivial solutions to homogeneous systems or non-zero determinants for square matrices. This theory operates within the subfield of Linear Algebra and Matrix Theory, relying strictly on formal definitions involving span, rank-nullity theorem applications, and basis set constructions. It serves as a foundational axiom distinguishing between unique solvability in direct systems and infinite solution sets, providing the structural criteria required before addressing inconsistent system resolution via orthogonal projection methods.
Prerequisite chain context: requires Homogeneous Systems of Linear Equations in Math.