 TOPICS # Linear System of Equations

A linear system of equations is a set of linear equations in variables (sometimes called "unknowns"). Linear systems can be represented in matrix form as the matrix equation (1)

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

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

If and the matrix is nonsingular, then the system has a unique solution in the variables. In particular, as shown by Cramer's rule, there is a unique solution if has a matrix inverse . In this case, (2)

If , then the solution is simply . If has no matrix inverse, then the solution set is the translate of a subspace of dimension less than or the empty set.

If two equations are multiples of each other, solutions are of the form (3)

for a real number. More generally, if , 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 components can be solved in terms of the last components to find the solution space.

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

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## References

Calc101.com. "Step-by-Step Linear Equations, Matrices and Determinants." http://calc101.com/webMathematica/matrix-algebra.jsp.

## Referenced on Wolfram|Alpha

Linear System of Equations

## Cite this as:

Weisstein, Eric W. "Linear System of Equations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LinearSystemofEquations.html