[1) Consider the following system of linear equations C+ += =1 " + (b- 1 )v + 2: =1 where a and b are parameters. (a) Determine, according to the values of a and b, whether the system is compatible, incompatible, determi nate or Indeterminate. (h) Solve the system for the values of a and b for which the system is compatible indeterminate.2 (1) Consider the following system of linear equations, 1-1- b =1 r+(a- D)y =b+1 i+ (a - D)y + (a - b)= =25+ 1 where a, b E R are parameters. (a) Classify the system according to the values of the parameters a, b. (b) Solve the above system for the values of a and b for which the system is underdetermined. How many parameters are needed to describe the solution?(2) The matrix 13 -1 A= 8 en 19 has A1 = 6 and Ag = 8 as eigenvalues. (You do not need to prove this). Solve the following. (a) Find the eigenvectors of the matrix A. (b) Justify whether the matrix A is diagonalizable. And, if so, find two matrices D and P such that A = PDP-1. (e) Find a matrix B such that B2 = A. It is enough to write B as the product of three matrices.(3) Given the linear map / : R' - R', /(zy.= ) = (x-y+=-t,-s+y-=+1) (a) Compute the dimensions of the kernel and the image and find some equations defining these subspaces. (b) Find a basis of the image of f and a basis of the kernel of f.(4) Given the set A= ((ry)ER' : Wysz, asi) (a) Draw the set A, computing its vertices. Draw its boundary and interior and discuss whether the set A is open, closed, bounded, compact and/or convex. You must explain your answer. (b) Consider the function f(x, y) = (2 - 1)2 + (y - 1)2 At what point(s) does the function / fail to be continuous? Determine if f attains a maximum and a minimum on the set A. State the theorems that you are using. (c) Draw the level curves of the function g(z, y) = (x - 2) + y' and use them to determine the maxima and minima of g on A.(5) Consider the function / : R' _ R if (x, y) * (0,0). if (x. y) = (0,0). (a) Study if the function f is continuous at the point (0,0). Study at which points of R? the function f is continuous. (b) Compute the partial derivatives of f at the point (0,0). (c) At which points of R" is the function f differentiable?(6) Given the quadratic form Q(z, v.=) = 2x' + y3 + 323 + 4ary + 2yz (a) Determine the matrix associated to the above quadratic form. (b) Classify the quadratic form, according to the values of the parameter a.(7) Consider the function f(r.y) = x' - In(x]) - 4 In(y?) + y (a) Compute the gradient vector and the Hessian matrix of f at any point (r, y) in domain of the function. (b) Determine the critical points of f and classify them. (c) Determine if the function f attains any extreme points on the set A= ((x,y) ER' :> >0. y>0)