Gausseliminációként

Gausseliminációként is the Hungarian term for Gaussian elimination, a fundamental algorithm in linear algebra used to solve systems of linear equations and to find the rank of a matrix. It systematically transforms a given matrix into row echelon form or reduced row echelon form through a series of elementary row operations. These operations include swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another row.

The primary goal of Gausseliminációként is to introduce zeros below the leading non-zero entries (pivots) in each row. Once the matrix is in row echelon form, the system of linear equations represented by the matrix can be solved more easily using back-substitution. If the process continues to achieve reduced row echelon form, where each pivot is 1 and all other entries in its column are zero, the solution to the system becomes immediately apparent.

Beyond solving linear systems, Gausseliminációként has applications in determining the invertibility of a matrix, calculating the determinant of a matrix, and finding the basis of vector spaces. It is a cornerstone of numerical linear algebra and a critical tool for understanding the properties and behavior of matrices and the systems they represent. The efficiency and generality of the algorithm make it indispensable in various scientific and engineering disciplines.