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Subject Item
dbr:Inverse_problem
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dbr:Unisolvent_functions
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dbr:List_of_numerical_analysis_topics
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dbr:Unisolvent_functions
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dbr:Unisolvent_functions
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Unisolvent functions
rdfs:comment
In mathematics, a set of n functions f1, f2, ..., fn is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors 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 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 unisolv
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dbr:Matrix_(mathematics) dbr:Vector_space dbr:Vector_(mathematics) dbr:Interpolation dbc:Interpolation dbr:Linearly_independent dbr:Finite_element_method dbr:Euclidean_space dbr:Unisolvence_theorem dbc:Approximation_theory dbr:Philip_J._Davis dbc:Numerical_analysis dbr:Polynomial dbr:Determinant dbr:Function_(mathematics) dbr:Domain_of_a_function dbc:Inverse_problems dbr:Polynomial_interpolation dbr:Basis_(linear_algebra) dbr:Linear_inverse_problem dbr:Inverse_problem dbr:Intermediate_value_theorem
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In mathematics, a set of n functions f1, f2, ..., fn is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors 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 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 (which form a vector space of dimension ) are unisolvent by the unisolvence theorem.
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