} _{j})=\sum _{i=1}^{m}a_{i,j}\mathbf {w} _{i}\qquad {\mbox{for }}j=1,\ldots ,n.}In other words, column j of A expresses the image of vj in terms of the basis vectors wi of W; thus this relation uniquely determines the entries of the matrix A. Note that the matrix depends on the choice of the bases: different choices of bases give rise to different, but equivalent matrices.[45] Many of the above concrete notions can be reinterpreted in this light, for example, the transpose matrix AT describes the transpose of the linear map given by A, with respect to the dual bases.[46]More generally, the set of m×n matrices can be used to represent the R-linear maps between the free modules Rm and Rn for an arbitrary ring R with unity.  When n = m composition of these maps is possible, and this gives rise to the matrix ring of n×n matrices representing the endomorphism ring of Rn.න්‍යාස කාණ්ඩ[සංස්කරණය]ප්‍රධාන ලිපිය: Matrix groupA group is a mathematical structure consisting of a set of objec