(1) Consider the following system of linear equations, a+7+0=-1 where a E R is a parameter. (a) Classify the system according to the values of a. (b) Solve the above system for the values of a for which the system is compatible indeterminate. (2) Given the matrix 1 = 2 - 1 where o, B E R are parameters, (a) Find the characteristic polynomial and the eigenvalues. (b) Determine for which values of the parameters o, S E R, the matrix is diagonalizable. (c) For the values of the parameters a = -1 and A = -8, find the corresponding diagonal matrix and the matrix change of basis. (3) Given the linear mapping / : R' - R. / (x. 17, 2, 1) = ( -5 -17-2,8 -9-1, -3x -3y -3:) (a) Compute the dimensions of the kernel and the image and a set of equations for these subspaces, (b) Find a basis of the image of / and a basis of the kernel of f. (4) Given the set (a) Draw the set S, its boundary and interior and discuss whether the set S is open, closed, bounded, compact and for convex. You must explain your answer. (b) Show that the function /(x, v) = (x - 1)' + (y - 1) has a maximum and a minimum on the set S, (c) Draw the level curves of /(a, y) and determine where the maxima and the minima of / on S. (5) Consider the function / ; RI - R / (x, ) = 347 i (x, V) # (0,0), if ( x, v) = (0,0). (a) Study If the function / is continuous at the point (0,0). (b) Compute the partial derivatives of f at the point (0,0). c) Determine at which points of R' the partial derivatives of / are continuous. (6) Consider the function ((a, v. =) =x' tay' + s' + 2any + 202 - 2ym [a) Find the Hessian matrix of f. (b) Study for which values of parameter a, the function / is strictly concave or strictly convex. (7) Consider the function /(x, y) = 8x + 2cy - 3x' + y' + 1. (a) Find the critical points of /. (b) Classify the critical points of / that you found in the previous part. (c) Determine whether / has any global extreme points on the set A = ((x, v) ER' -