A linear systems calculator is a useful tool for finding solutions to systems of linear equations. A system of linear equations is a set of linear equations that must be solved simultaneously. The solutions of this system can be found using several methods, such as Gauss elimination, Gauss-Jordan elimination, Cramer's rule, among others.
Remembering that the equations must be separated by a line and the coefficients and the independent term must be separated by a space. By clicking on “Calculate”, the calculator will solve the system of equations using the Gauss-Jordan elimination method and display the solutions in the results table.
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Online Linear Systems Calculator
About Linear Systems
A linear system is a set of linear equations that must be solved simultaneously to determine the values of the variables involved. Linear equations are first-degree equations with one or more variables, such as:
2x + 3y - z = 7 x - 4y + 5z = 10 3x + y + z = 6
These equations can be written in the matrix form Ax = b, where A is the matrix of coefficients, x is the vector of variables and b is the vector of constants. In the example above, we have:
A = [2 3 -1; 1 -4 5; 3 1 1] x = [x; y; z] b = [7; 10; 6]
Solving a linear system means finding the values of variables x, y and z that satisfy all equations simultaneously. There are several techniques to solve linear systems, such as Gauss elimination, LU decomposition, Cholesky decomposition, among others.
Linear systems have many applications in science, engineering, economics and many other areas. They can be used, for example, to model physical systems such as electrical circuits, mechanical systems and thermal systems. They are also used in data analysis, optimization, and other areas of applied mathematics.
Methods of Calculating Linear Systems
Below is a brief description of each of the common methods of calculating linear systems:
- Gauss elimination: also known as the Gauss-Jordan elimination method, this method involves transforming the matrix of coefficients into an upper (or lower) triangular matrix through elementary row operations. Once the matrix is triangularized, the system can be easily solved using the substitution method.
- LU decomposition: this method involves factoring the matrix of coefficients into two triangular matrices (one lower and one upper), through elementary row operations. The matrix of coefficients is factored using elementary row operations, and then the system can be easily solved using the factored matrices.
- Cholesky Decomposition: this method is a specialized form of LU decomposition for symmetric and positive definite matrices. It involves factoring the matrix of coefficients into a lower triangular matrix and its conjugate transpose, and then solving the resulting triangular systems.
- Inverse matrix method: this method involves inverting the matrix of coefficients and multiplying the result by the vector of constants. Although a straightforward method, it can be computationally expensive for large matrices.
- Jacobi's method: this method is an iterative method that involves decomposing the matrix of coefficients into a diagonal matrix and their differences. It requires that the matrix of coefficients be diagonally dominant.
- Gauss-Seidel method: this method is another iterative method that involves decomposing the matrix of coefficients into a lower triangular matrix and their differences. It is similar to Jacobi's method, but it can converge faster for some matrices.
- Successive relaxation method: this method is an extension of the Gauss-Seidel method that involves adding a relaxation parameter to speed up convergence. It can be especially useful for ill-conditioned matrices.
- Conjugate iteration method: this method is an iterative method that is especially useful for symmetric and positive definite matrices. It involves choosing a set of orthogonal search vectors that are iteratively refined to converge to the solution.
- Krylov's method: this method is another iterative method that uses the idea of Krylov spaces, which are subspaces generated by applying the matrix of coefficients to a vector. The method involves projecting the vector of constants onto Krylov space and minimizing the approximation error.
- QR decomposition method: this method involves factoring the matrix of coefficients into an orthogonal matrix and an upper triangular matrix. It can be used to solve linear systems and to calculate matrix eigenvalues and eigenvectors.
