Bachelor Degree in Informatics Engineering Barcelona School of Informatics Mathematics 1 Part II: Linear Algebra Exercises and problems Academic Year 2019-2020(1) Departament de Matem`atiques Universitat Polit`ecnicade Catalunya The problems of this collection were initially gathered by Anna de Mier and Montserrat Mau- reso. Many of them were taken from the problem sets of several courses taught over the years by the members of the Departament de Matem`aticaAplicada 2. Other exercises came from the bibliography of the course or from other texts, and some of them were new. Since Mathematics 1 was first taught in 2010 several problems have been modified or rewritten by the professors involved in the teaching of the course. We would like to acknowledge the assistance of the scholar Gabriel Bernardino in the writing of the solutions. Translation by Josep Elgueta and the scholar Bernat Coma. This problem list has been revised during the academic year 2018/19. Contents 5 Matrices, systems of linear equations and determinants 1 5.1 Matrix algebra :::::::::::::::::::::::::::::::::::::: 1 5.2 Systems of linear equations :::::::::::::::::::::::::::::: 4 5.3 Determinants :::::::::::::::::::::::::::::::::::::::: 6 6 Vector spaces 7 7 Linear maps 15 8 Diagonalization 23 Review exercises 26 5 Matrices, systems of linear equations and determinants Unless otherwise indicated, we always work in the field R of real numbers. 5.1 Matrix algebra 5.1 Given the matrices 0 2 1 0 1 0 1 2 4 1 0 2 0 01 1 0 3 A = −3 0 −1 ;B = 1 −4 3 ;C = B C @ A @ A B−1 0 2 C 2 1 2 −1 3 2 @ A 4 5 −1 compute: 1) 3A; 2) 3A − B; 3) AB; 4) BA; 5) C(3A − 2B). 0 2 1 0 2 1 5.2 Compute the products (1 2 − 3) @ 1 A and @ 1 A (1 2 − 3): 5 5 5.3 Let A and B be matrices such that AB is a square matrix. Show that the product BA is well defined. 5.4 For the following matrices A and B, give the elements c13 and c22 of the matrix C = AB without computing all the elements of C. 0 2 0 01 1 2 1 A = ;B = 1 −4 3 : −3 0 −1 @ A −1 3 2 5.5 A company makes bags and suitcases in two different factories. The table below gives the cost, in thousands of euros, of manufacturing each product in each factory: Factory 1 Factory 2 Bags 135 150 Suitcases 627 681 2 Chapter 5. Matrices, systems of linear equations and determinants Answer the following questions using matrix operations. 1) Knowing that the personnel costs represent 2/3 of the total cost, find the matrix that gives the personnel cost of each product in each factory. 2) Find the matrix that gives the cost of the material of each product in each factory, assuming that, in addition to the personnel and material costs, there is a cost of 20.000 euros for each product in each factory. 5.6 In this exercise we want to find a formula giving the successive powers of a diagonal matrix. a) Compute A2, A3 and A5 if 02 0 01 A = @0 −1 0A : 0 0 3 b) Compute A32. c) Let D be a diagonal n × n matrix whose diagonal entries are λ1; λ2; : : : ; λn. Make a guess about Dr, for r 2 Z, r ≥ 1, and prove it by induction. 1 −2 −2 1 5.7 Let A = and B = . Compute (AB)t and BtAt. Observe that AB −2 3 1 1 can be a non-symmetric matrix even when both A and B are symmetric. 5.8 Give an example of two 2 × 2 matrices A and B such that (AB)t 6= AtBt. 5.9 Let I be the identity matrix and O the null matrix of M2×2(R). Find matrices A; B; C; D; E 2 M2×2(R) such that 1) A2 = I and A 6= I; 3) C2 = C and C 6= I; O; 2) B2 = O and B 6= O; 4) DE = O but E 6= D and ED 6= O. 5.10 Let A and B be two symmetric matrices of the same type. Prove that AB is a symmetric matrix if and only if A and B commute. 5.11 Do the following equalities hold for all matrices A; B 2 Mn(R)? If not, give a condition on A and B ensuring that the equations hold. 1)( A + B)2 = A2 + B2 + 2AB; 2)( A − B)(A + B) = A2 − B2: 5.1. Matrix algebra 3 5.12 Let A and B be square matrices of the same type. A is said to be similar to B if there exists an invertible matrix P such that B = P −1AP . If A is similar to B, prove: 1) B is similar to A. In general we say that A and B are similar. 2) The relation \being similar" is an equivalence relation. 3) A is invertible if and only if B is invertible. 4) At is similar to Bt. 5) If An = O and C is an invertible matrix of the same type as A, then (C−1AC)n = O. 5.13 Find a matrix in row echelon form equivalent to each of the following matrices. Give the rank of the matrix in each case. 05 11 61 0 0 1 2 3 41 0 1 0 2 31 0−3 11 2 1 4 −1 0 1 2 3 1) 2 1 1 3 2) 2 0 3) B C 4) B C @ A @ A B3 −2 8C B−2 −1 0 1 2C −1 2 0 0 6 4 @ A @ A 0 0 4 −3 −2 −1 0 1 5.14 Find the inverse of the following elementary matrices. 0 1 00 0 11 0k 0 01 1) 1 0 3) @0 1 0A 5) @0 1 0A, k 6= 0 1 0 0 0 0 1 01 0 01 5 0 4) 0 1 0 2) @ A 0 1 0 −3 1 5.15 Using the Gauss-Jordan method, find, if it exists, the inverse of each of the following matrices. 1 2 0 3 1 5 1 0−2 3 −1 −11 1) 3 5 2) @ 2 4 1 A B−1 1 1 1 C 3) B C −4 2 −9 @−1 −1 1 2 A 3 1 −2 −4 4) A = (ai;j)4×4; such that ai;j = 1 if ji − jj ≤ 1; and ai;j = 0 otherwise. j−1 5) A = (ai;j)4×4; such that ai;j = 2 if i ≥ j; and ai;j = 0 otherwise. 6) A = (ai;j)4×4; such that ai;i = k; ai;j = 1 if i − j = 1; and ai;j = 0 otherwise. 4 Chapter 5. Matrices, systems of linear equations and determinants 5.2 Systems of linear equations 5.16 Which of the following equations are linear in x, y and z? 1) x + 3xy + 2z = 2; 3) x − 4y + 3z1=2 = 0; 5) z + x − y−1 + 4 = 0; p 2 π 2) y + x + 2z = e ; 4) y = z sin 4 − 2y + 3; 6) x = z. 5.17 Find a system of linear equations for each of the following augmented matrices. 01 0 3 21 1 2 3 4 5 3) 1) @2 1 1 3A 1=3 1=4 1=5 1=2 1 0 −1 2 4 01 0 0 0 11 0 1 −1 5 −2 B0 1 0 0 2C 4) B C 2) @ 1 1 0 A @0 0 1 0 3A 1 −1 1 1 0 0 1 4 5.18 Answer the following questions. Justify your answers. 1) What is the rank of the system matrix of a determined consistent system with 5 equations and 4 unknowns? What about if the system is underdetermined? 2) How many equations are needed at least to have an underdetermined consistent system with 2 degrees of freedom and rank 3? How many unknowns would this system have? 3) Is it possible to have a determined consistent system with 7 equations and 10 unknowns? 4) Can a system with fewer equations than unknowns be inconsistent? 5) Give a determined consistent system, an underdetermined consistent system and an in- consistent system, each with 3 unknowns and 4 equations. 5.19 Solve the following systems of linear equations with coefficients in Z2. Use Gaussian elimination and give the solution in parametric form. 8 x + y = 1 8 x + y = 1 8 x + y = 0 < < < 1) x + z = 0 2) y + z = 1 3) y + z = 0 : x + y + z = 1 : x + z = 1 : x + z = 0 5.2. Systems of linear equations 5 5.20 Solve the following systems of linear equations. Use Gaussian elimination and give the solution in parametric form. 8 x + y + 2z = 8 8 x − y + 2z − w = −1 > > <> −x − 2y + 3z = 1 <> 2x + y − 2z − 2w = −2 1) 3) 3x − 7y + 4z = 10 −x + 2y − 4z + w = 1 > > : 3y − 2z = −1 : 3x − 3w = −3 8 x − y + 2z = 3 8 x + 3x − 2x + 2x = 0 > > 1 2 3 5 <> 2x − 2y + 5z = 4 <> 2x + 6x − 5x − 2x + 4x − 3x = −1 2) 4) 1 2 3 4 5 6 x + 2y − z = −3 5x + 10x + 15x = 5 > > 3 4 6 : 2y + 2z = 1 : 2x1 + 6x2 + 8x4 + 4x5 + 18x6 = 6 5.21 Solve the following homogeneous systems of linear equations. Use Gaussian elimination and give the solution in parametric form. 8 2x + 2y + 2z = 0 8 x − 3x + x = 0 < < 2 3 4 1) −2x + 5y + 2z = 0 3) x1 + x2 − x3 + 4x4 = 0 : −7x + 7y + z = 0 : −2x1 − 2x2 + 2x3 − 8x4 = 0 8 2x − 4y + z + w = 0 > 8 2x + 2x − x + x = 0 > x − 5y + 2z = 0 > 1 2 3 5 <> <> −x − x + 2x − 3x + x = 0 2) −2y − 2z − w = 0 4) 1 2 3 4 5 x + x + x + 2x = 0 > x + 3y + w = 0 > 1 2 3 5 > : 2x + 2x + 2x = 0 : x − 2y − z + w = 0 3 4 5 5.22 Discus the following systems of linear equations according to the values of the parameters (assumed to be real).