transposeMatrix

The transpose of a matrix is a fundamental operation in linear algebra. To transpose a matrix means to interchange its rows and columns. If a matrix A has dimensions m x n, meaning it has m rows and n columns, its transpose, denoted as A^T or A', will have dimensions n x m. Each element A[i][j] in the original matrix becomes the element A^T[j][i] in the transposed matrix. In simpler terms, the element in the i-th row and j-th column of the original matrix moves to the j-th row and i-th column of the transposed matrix.

For example, consider a 2x3 matrix:

1 2 3

4 5 6

Its transpose would be a 3x2 matrix:

1 4

2 5

3 6

The operation of transposing a matrix has several useful properties. For instance, the transpose of a transpose is the original matrix: (A^T)^T = A. Also, the transpose of a sum of matrices is the sum of their transposes: (A + B)^T = A^T + B^T. Furthermore, for scalar multiplication, (kA)^T = kA^T. In matrix multiplication, the transpose of a product of matrices is the product of their transposes in reverse order: (AB)^T = B^T A^T. The transpose operation is widely used in various mathematical fields, including calculus, statistics, and computer graphics, particularly in solving systems of linear equations and in the study of vector spaces.