Transponeerimine

Transponeerimine is a mathematical operation performed on a matrix. It involves interchanging the rows and columns of the matrix. If a matrix A has dimensions m x n, its transpose, denoted as A^T, will have dimensions n x m. The element in the i-th row and j-th column of the original matrix A becomes the element in the j-th row and i-th column of the transposed matrix A^T. In other words, A^T_{ij} = A_{ji}.

For example, if matrix A is [[1, 2], [3, 4], [5, 6]], then its transpose A^T would be [[1, 3, 5], [2, 4, 6]]. Transponeerimine has several important properties. The transpose of a transposed matrix is the original matrix: (A^T)^T = A. The transpose of a sum of matrices is the sum of their transposes: (A + B)^T = A^T + B^T. The transpose of a product of matrices is the product of their transposes in reverse order: (AB)^T = B^T A^T. This operation is fundamental in various areas of linear algebra, including solving systems of linear equations, calculating determinants, and in the study of vector spaces and transformations.