Editing Determinant (section)

=== Linear independence ===
Determinants can be used to characterize [[linear independence|linearly dependent]] vectors: <math>\det A</math> is zero if and only if the column vectors of the matrix <math>A</math> are linearly dependent.<ref>{{harvnb|Lang|1985|loc=§VII.3}}</ref> For example, given two linearly independent vectors <math>v_1, v_2 \in \mathbf R^3</math>, a third vector <math>v_3</math> lies in the [[Plane (geometry)|plane]] [[Linear span|spanned]] by the former two vectors exactly if the determinant of the <math>3 \times 3</math> matrix consisting of the three vectors is zero. The same idea is also used in the theory of [[differential equation]]s: given functions <math>f_1(x), \dots, f_n(x)</math> (supposed to be <math>n-1</math> times [[differentiable function|differentiable]]), the [[Wronskian]] is defined to be
:<math>W(f_1, \ldots, f_n)(x) =
  \begin{vmatrix}
            f_1(x) &         f_2(x) & \cdots &         f_n(x) \\
           f_1'(x) &        f_2'(x) & \cdots &        f_n'(x) \\
            \vdots &         \vdots & \ddots &         \vdots \\
    f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)
\end{vmatrix}.</math>

It is non-zero (for some <math>x</math>) in a specified interval if and only if the given functions and all their derivatives up to order <math>n-1</math> are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of [[analytic function]]s, this implies the given functions are linearly dependent. See [[Wronskian#The Wronskian and linear independence|the Wronskian and linear independence]]. Another such use of the determinant is the [[resultant]], which gives a criterion when two [[polynomial]]s have a common [[root of a function|root]].<ref>{{harvnb|Lang|2002|loc=§IV.8}}</ref>