Maatriksivorm

Maatriksivorm, or matrix form, is a concept in linear algebra used in Estonian mathematics to describe standard representations of a matrix obtained through row operations to reveal structure and simplify computations. The most widely used forms are row echelon form (REF) and reduced row echelon form (RREF).

These forms are reached by elementary row operations: swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. Two matrices are considered row-equivalent if one can be transformed into the other by such operations. Row echelon form requires that all nonzero rows appear above any zero rows; the leading coefficient (the first nonzero entry) of a nonzero row is to the right of the leading coefficient of the row above it; all entries in a column below a leading coefficient are zero. Reduced row echelon form adds that each leading coefficient is 1 and that it is the only nonzero entry in its column.

Existence and uniqueness: every matrix over a field has a row-equivalent REF, and the reduced row echelon form is unique for a given matrix. Gaussian elimination is the standard method to obtain REF, while Gauss–Jordan elimination yields the RREF.

Applications: maatriksivorm is used to solve linear systems, determine the rank of a matrix, and identify dependencies among rows or columns. The pivot positions in REF or RREF reveal the dimension of the solution space and the consistency of linear systems.

Example: consider A = [ [1, 2, 3], [4, 5, 6] ]. Its REF is [ [1, 2, 3], [0, -3, -6] ], and its RREF is [ [1, 0, -1], [0, 1, 2] ].