Systems of Linear Equations
The program on this page uses Gaussian elimination to solve any system of linear equations of the form
A x = b.
A is the matrix of coefficients, x is the vector of unknowns and b is the vector of the right-hand side.
The system of equations can be underdetermined. A may be of any decline of rank.
A unique solution only exists for regular matrix A, that is if det(A)≠0.
If det(A)=0 and rk(A)<rk(A|b), then there is no solution.
A|b is the so called augmented coeffient matrix. A is augmented by the vector b.
If det(A)=0 and rk(A)=rk(A|b), an infinitive number of solutions exist.
These solutions can be presented in the form of a general solution.
The general solution is the sum of a particular solution of the inhomogeneous system of equations
and the linear combination of all solutions of the homogeneous system of equations formed with the help of free parameters λi (λi ∈ ℝ).
The number of necessary parameters equals the difference between the number of unknowns and the rank of the system matrix.
The particular solution presented here is orthogonal to the set of solutions of the homogeneous system.
It is possible to define auxiliary parameters, that can be used in the system matrix and the vector of the right-hand side.
Unused cells are interpreted as 0, so there is no need to specify zeros.
You can use cursor keys to change the input cell. CR initiates a new calculation.
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