Grammian determinant

The Grammian determinant provides a necessary and sufficient method of determining whether a set of continuous functions ${f_{1},f_{2},\\ldots,f_{n}}$ is linearly independent on an interval $I=[a,b]$ with respect to the inner product

\\langle f_{i}|f_{j}\\rangle=\\int_{I}f_{i}f_{j}

It is defined as:

G(f_{1},f_{2},\\ldots,f_{n})=\\left\\lvert\\begin{array}[]{@{}cccc@{}}\\int_{I}(f_{%1})^{2}&\\int_{I}f_{1}f_{2}&\\cdots&\\int_{I}f_{1}f_{n}\\\\\\int_{I}f_{2}f_{1}&\\int_{I}(f_{2})^{2}&\\cdots&\\int_{I}f_{2}f_{n}\\\\\\vdots&\\vdots&\\ddots&\\vdots\\\\\\int_{I}f_{n}f_{1}&\\int_{I}f_{n}f_{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.