information:classification, structural:summarization, structural:reformatting
Linear independence

Determinants can be used to characterize linearly dependent vectors: det A {\displaystyle \det A} is zero if and only if the column vectors (or, equivalently, the row vectors) of the matrix A {\displaystyle A} are linearly dependent. For example, given two linearly independent vectors v 1, v 2 ∈ R 3 {\displaystyle v_{1},v_{2}\in \mathbf {R} ^{3}}, a third vector v 3 {\displaystyle v_{3}} lies in the plane spanned by the former two vectors exactly if the determinant of the 3 × 3 {\displaystyle 3\times 3} -matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given functions f 1 (x), …, f n (x) {\displaystyle f_{1}(x),\dots,f_{n}(x)} (supposed to be n − 1 {\displaystyle n-1} times differentiable), the Wronskian is defined to be

It is non-zero (for some x {\displaystyle x}) in a specified interval if and only if the given functions and all their derivatives up to order n − 1 {\displaystyle n-1} are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence. Another such use of the determinant is the resultant, which gives a criterion when two polynomials have a common root.