# Image transcription text [1) Consider the following system of linear equations C+ += =1 " + (b- 1 )v + 2: =1 where a and

Image transcription text [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. Image transcription text 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? Image transcription text (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. Image transcription text (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. Image transcription text (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. Image transcription text (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? Image transcription text (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. Image transcription text (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)

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