linear independence

No vector in the set can be built from the others; only the trivial combination gives $\mathbf 0$ .

Vectors $\mathbf v_1,\dots,\mathbf v_k$ are linearly independent if the only solution to $c_1\mathbf v_1+\cdots+c_k\mathbf v_k=\mathbf 0$ is the trivial one, $c_1=\cdots=c_k=0$ . If some other combination — not all zero — also gives $\mathbf 0$ , the set is linearly dependent, and that nontrivial equation is a linear dependence relation. Testing independence always reduces to solving a homogeneous linear system in the coefficients.