A set of vectors in a vector space is defined as linearly independent if and only if the homogeneous linear combination equation yields solely the trivial zero solution, thereby precluding any non-trivial dependency relations among the elements. Conversely, a set is classified as linearly dependent if at least one vector can be expressed as a linear combination of the others or if the associated coefficient matrix possesses free variables during row reduction. This concept serves as a fundamental determinant for assessing the dimensionality and structural integrity of subspaces within the broader discipline of linear algebra.
Prerequisite chain context: requires Systems of Linear Equations Representation.