The Grammian determinant provides a necessary and sufficient method of determining whether a set of continuous functions ${f_1, f_2, \dotsc, f_n}$ is linearly independent on an interval $I = [a, b]$ with respect to the inner product $$\langle f_i | f_j\rangle = \int_I f_if_j$$ It is defined as: $$ G(f_1, f_2, \dotsc, f_n) = \left\lvert\begin{array}{@{}cccc@{}} \int_I (f_1)^2 & \int_I f_1f_2 & \cdots & \int_I f_1f_n\\ \int_I f_2f_1 & \int_I (f_2)^2 & \cdots & \int_I f_2f_n\\ \vdots & \vdots & \ddots & \vdots\\ \int_I f_nf_1 & \int_I f_nf_2 & \cdots & \int_I (f_n)^2\\ \end{array}\right\rvert $$ If the functions are continuous on $I$ then $G = 0$ if and only if the set of functions is linearly dependent. Note that the Grammian determinant is a special case of the more general Gram determinant.