Mathematics IB Tutorial 6 (week 7) Semester 2, 2016 1. Prove that a linear transformation F : Rn Rn is invertible if and only if ker F = {0}. 8x2 . We can easily see that it has 3x2 + 1 a single root at x = 0, however it is interesting to see how Newton's Method goes at trying to find this root. Find the formula for xn+1 in terms of xn which gives successive approximations of the root of this function using Newton's method. Apply this formula for each of the following initial values: x0 = 1, x0 = 0.5, x0 = 2, and in each case find all xn up to x5 . (If the numbers get very large or small you can just do a rough order of magnitude calculation). 2. Consider the function f (x) = Explain the differing outcomes in terms of the graph of the function shown below: 3 2 1 2 1 0 1 2 1