[1) Consider the following system of linear equations, a toy + = =0 at/+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 where a, B E R are parameters, (a) Find the characteristic polynomial and the eigenvalues. (b) Determine for which values of the parameters a, & E R, the matrix is diagonalizable. c) For the values of the parameters a = -1 and S = -8, find the corresponding diagonal matrix and the matrix change of basis . 3) Given the linear mapping f : R' - R f(x. M. =,( ) = (-4 -1 -2,8 -y-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 f 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 /or convex. You must explain your answer. (b) Show that the function /(x, y) = (x - 1)' + (y - 1)' has a maximum and a minimum on the set S. (c) Draw the level curves of /(x, y) and determine where the maxima and the minima of f on S. (5) Consider the function / : R? - R 1 (x. v) = Says if (a, v) # (0,0 ), if ( x, v) = (0,0). (a) Study if the function f 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. G) Consider the function f(x, v, = ) =x' tay' + s' + 2ary + 2re - 2y= (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 f(x,y) = 8x3 + 2ry -3x* + y' + 1. (a) Find the critical points of f. (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' : -