ධාන ලිපියන්: Linear transformation  සහ Transformation matrixThe vectors represented by a 2-by-2 matrix correspond to the sides of a unit square transformed into a parallelogram.Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps. A real m-by-n matrix A gives rise to a linear transformation Rn → Rm mapping each vector x in Rn to the (matrix) product Ax, which is a vector in Rm. Conversely, each linear transformation f: Rn → Rm arises from a unique m-by-n matrix A: explicitly, the (i, j)-entry of A is the ith coordinate of f(ej), where ej = (0,...,0,1,0,...,0) is the unit vector with 1 in the jth position and 0 elsewhere. The matrix A is said to represent the linear map f, and A is called the transformation matrix of f.For example, the 2×2 matrixA=[acbd]{\displaystyle \mathbf {A} ={\begin{bmatrix}a&c\\b&d\end{bmatrix}}\,}can be viewed as the transform of the unit s