4. Sharpness of condition number estimates [4pt] Let A E R^n x n be invertible. Let b E R^n\{0},and Ax = b, Ax' = b' and denote the perturbations by

Ab = b' – b and Ax = x' – x . Show that the inequality obtained in Theorem 2.11 is sharp. That is, find vectors b, Ab for which \frac{\|\Delta x\|_{2}}{\|x\|_{2}}=\kappa_{2}(A) \frac{\|\Delta b\|_{2}}{\|b\|_{2}} where k2(A) is the condition number of A under the 2-norm. (Hint: consider the eigenvectors of A^T A.)