The Rotation of a Coordinate System as a Linear Transformation

and call the column vector c the representation of the vector x with respect to the basis X . Now, let V and W be vector spaces over F , let X = (x,x, . . . ,x) be a basis of V , let Y = (y,y, . . . , y) be a basis of W , and let T : V → W be a linear transformation. For any i with 1 ≤ i ≤ n we have Tx = ∑m j=1 y aji with some aji ∈ F . For an arbitrary vector x = V such that c = (c1, . . . , cn) = RXx we have x = ∑n i=1 x ci and Tx = ∑i j=1 y j ∑n i=1 ajici. In matrix form, this equation will be written as Tx = TXc = YAc where A is the n×m matrix with entries aji in the jth row and the ith column. In the last equation, we may omit the column vector c on the right-hand side and write TX = YA. We will call the matrix A the representation of the linear transformation T with respect to the bases X and Y, and write