Lahendusruumi

Lahendusruum, in mathematics often translated as the solution space, denotes the set of all solutions to a problem, most commonly to a system of linear equations or to a differential equation. For a linear system Ax = b, the solution set S = { x in F^n | Ax = b } is either empty or an affine subspace of the ambient vector space. If there is at least one solution, S can be written as x_p + N(A), where x_p is a particular solution and N(A) = { x | Ax = 0 } is the null space of A. The system is consistent iff b lies in the column space of A. The dimension of S, when nonempty, equals the nullity of A, which is n - rank(A). In the purely homogeneous case b = 0, S = N(A) is a vector subspace.

Computation: S can be found by row reducing [A|b] to echelon form or reduced row echelon form, solving for pivot and free variables. A basis for N(A) is obtained from the free variable expressions, and a particular solution x_p can be found by setting free variables to zero or by other means.

Applications: The concept is central in linear algebra, systems theory, and differential equations. In differential equations, the solution space of a homogeneous linear ODE or system forms a vector space, and the general solution of an inhomogeneous equation is a particular solution plus the solution space of the corresponding homogeneous equation.

Origin and usage: The term lahendusruum reflects Estonian mathematical terminology and is used in education and literature to describe the set of all solutions to mathematical problems.