I just found the determinant of a matrix and it was 0. I know this means that the linear transformation from which the matrix was derived isn't isomorphic. Why is this, though?
That means the standard matrix, A, is not invertible. Which means what? Let b be a coloum vector. The equation Ax=b either has no solutions or infinitely many. In either case the solutions for x IS NOT unique. Thus, the map cannot be both one-to-one and onto.