What have I done??? 1. The problem statement, all variables and given/known data (linear transformation) Let T: R2 -> R3 be a linear transformation such that T(e1) = u1 and T(e2) = u2, where u1 = [1; 0; -1] and u2 = [2; 1; 0]. Find each of the following: T([1; 1]) and T([2; -1]) 2. Relevant equations 3. The attempt at a solution Here is the thing. I've been sitting with this problem for a good hour, and cannot figure out. I just started playing around with numbers, toss em around a little bit, and in the end I've got the correct answer... But can somebody explain me, WHY is this the right way of doing it?? lol First, I looked at u1 and u2 as a matrix A = [1 2; 0 1; -1 0]. Then i did a familiar A*e1 = b1 and A*e2 = b2, where I got b1= [1; 0; -1] and b2 = [2; 1; 0].. (thinking out loud, every matrix multiplication is a transformation, so it must be relating somehow..) I let b1 and b2 be matrix (i call it transformation matrix) B = [1 2; 0 1; -1 0] then I decided to multiply this transformation matrix B by a vector that needs to be transformed from R2 to R3 [1; 1] and got [1 2; 0 1; -1 0] * [1; 1] = [3; 1; -1] By why in the name of Albert did it work?? lol did I have a correct logic through out the problem??