The quotient space R^n / U is isomorphic to the subspace W, where U is the kernel of the matrix A and W is the image of A. The linear map T_A : R^n -> R^m defined by T_A(X) = A_X establishes this relationship. The isomorphism theorem confirms that the image of A is isomorphic to the quotient space of R^n by its kernel, thus providing a clear mathematical framework for understanding this relationship.
PREREQUISITESMathematics students, particularly those studying linear algebra, educators teaching vector space concepts, and researchers exploring linear transformations and their properties.
crypt50 said:Let A be an m x n matrix with entries in R. Let T_A : R^n -> R^m be the linear map T_A(X) = A_X. Let U be the solution set of the homogeneous linear system A_X = O. Let W be the set of all vectors Y such that Y = A_X for some X in R^n. I don't really know what I'm supposed to do here, any help would be greatly appreciated.