Linear System of Equations

A linear system of equations is a set of n linear equations in k variables (sometimes called "unknowns"). Linear systems can be represented in matrix form as the matrix equation


where A is the matrix of coefficients, x is the column vector of variables, and b is the column vector of solutions.

If k<n, then the system is (in general) overdetermined and there is no solution.

If k=n and the matrix A is nonsingular, then the system has a unique solution in the n variables. In particular, as shown by Cramer's rule, there is a unique solution if A has a matrix inverse A^(-1). In this case,


If b=0, then the solution is simply x=0. If A has no matrix inverse, then the solution set is the translate of a subspace of dimension less than n or the empty set.

If two equations are multiples of each other, solutions are of the form


for t a real number. More generally, if k>n, then the system is underdetermined. In this case, elementary row and column operations can be used to solve the system as far as possible, then the first (k-n) components can be solved in terms of the last n components to find the solution space.

See also

Cramer's Rule, Determinant, Linear Equation, Matrix, Matrix Equation, Matrix Inverse, Null Space, Simultaneous Equations, System of Equations

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References "Step-by-Step Linear Equations, Matrices and Determinants."

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Linear System of Equations

Cite this as:

Weisstein, Eric W. "Linear System of Equations." From MathWorld--A Wolfram Web Resource.

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