1. The problem statement, all variables and given/known data Determine whether the following mappings f is onto or one-to-one. Is f an isomorphism? a) f maps R2 into R2 and is defined by f(x,y) = (x-2y, x+y) b) f maps R2 into R3 and is defined by f(x,y) = (x, y, x+y) i) f maps R3 into P2(R), defined by f(a1, a2, a3) = a2 - a3x + (1-a1)x^2 j) f maps Rmxn into Rnxm and is defined by f(A) = A^T (transpose of A) for all A in Rmxn 2. Relevant equations 3. The attempt at a solution Can I solve this by forming the matrix of the output? ie I have x-2y = 0 and x+y = 0 as a matrix and row reduce, I get a 2x2 identity matrix and therefore 1) There are always solution for any augmented matrix, so f is onto 2) There are no parameters in the matrix and both variables always have a unique solution for every augmented matrix, so f is one-to-one Is that right? For b) I get form the 3x2 matrix and row reduce, then a have third row of zero and no parameter, therefore f is not onto since there will always be some component of R3 that has no solution (hence the zero row). But f is 1-1 since both variables always have a unique solution for any augmented matrix (hence it has 2 leading ones). Does that explanation seem right? For i) I form the matrix and row reduce to a I3x3, therefore f is onto and 1-1. For j) f is one-to-one since it forms the transpose of A and every row of A is mapped to every column of the image once. It is onto since this applies to every matrix of Rmxn, therefore transposed into every matrix of Rnxm.