In mathematics, a set of n functions f1, f2, ..., fn is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors
[ f 1 ( x 1 ) f 1 ( x 2 ) ⋮ f 1 ( x n ) ] , [ f 2 ( x 1 ) f 2 ( x 2 ) ⋮ f 2 ( x n ) ] , … , [ f n ( x 1 ) f n ( x 2 ) ⋮ f n ( x n ) ] {\displaystyle {\begin{bmatrix}f_{1}(x_{1})\\f_{1}(x_{2})\\\vdots \\f_{1}(x_{n})\end{bmatrix}},{\begin{bmatrix}f_{2}(x_{1})\\f_{2}(x_{2})\\\vdots \\f_{2}(x_{n})\end{bmatrix}},\dots ,{\begin{bmatrix}f_{n}(x_{1})\\f_{n}(x_{2})\\\vdots \\f_{n}(x_{n})\end{bmatrix}}}
are linearly independent for any choice of n distinct points x1, x2 ... xn in Ω. Equivalently, the collection is unisolvent if the matrix F with entries fi(xj) has a nonzero determinant: det(F) ≠ 0 for any choice of distinct xj's in Ω. Unisolvency is a property of vector spaces, not just particular sets of functions. That is, a vector space of functions of dimension n is unisolvent if given any basis (equivalently, a linearly independent set of n functions), the basis is unisolvent (as a set of functions). This is because any two bases are related by an invertible matrix (the change of basis matrix), so one basis is unisolvent if and only if any other basis is unisolvent. Unisolvent systems of functions are widely used in interpolation since they guarantee a unique solution to the interpolation problem. The set of polynomials of degree at most d {\displaystyle d} (which form a vector space of dimension d + 1 {\displaystyle d+1} ) are unisolvent by the unisolvence theorem.
Examples 1, x, x2 is unisolvent on any interval by the unisolvence theorem 1, x2 is unisolvent on [0, 1], but not unisolvent on [−1, 1] 1, cos(x), cos(2x), ..., cos(nx), sin(x), sin(2x), ..., sin(nx) is unisolvent on [−π, π] Unisolvent functions are used in linear inverse problems.
Unisolvence in the finite element method When using "simple" functions to approximate an unknown function, such as in the finite element method, it is useful to consider a set of functionals { f i } i = 1 n {\displaystyle \{f_{i}\}_{i=1}^{n}} that act on a finite dimensional vector space V h {\displaystyle V_{h}} of functions, usually polynomials. Often, the functionals are given by evaluation at points in Euclidean space or some subset of it. For example, let
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