Mathematics 1. Part II: Linear Algebra. Exercises and problems

Bachelor Degree in Informatics Engineering Barcelona School of Informatics Mathematics Part II: Linear Algebra Eercises and problems February 5 Departament de Matemàtica Aplicada Universitat Politècnica de Catalunya

The problems of this collection were initially gathered by Anna de Mier and Montserrat Maureso. Many of them were taken from the problem sets of several courses taught over the years by the members of the Departament de Matemàtica Aplicada. Other eercises came from the bibliography of the course or from other tets, and some of them were new. Since Mathematics was first taught in 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.

Contents 5 Matrices, systems of linear equations and determinants 5. Matri algebra...................................... 5. Systems of linear equations.............................. 4 5.3 Determinants........................................ 6 6 Vector spaces 8 7 Linear maps 6 8 Diagonalization 3 Review eercises 7

5 Matrices, systems of linear equations and determinants Unless otherwise indicated, we always work in the field R of real numbers. 5. Matri algebra 5. Given the matrices 4 A = 3, B = 4 3, C = 3 compute: ) 3A; ) 3A B; 3) AB; 4) BA; 5) C(3A B). 5. Compute the products ( 3) 5 and 5 ( 3). 3 4 5 5.3 Let A and B be matrices such that AB is a square matri. Show that the product BA is well defined. 5.4 For the following matrices A and B, give the elements c 3 and c of the matri C = AB without computing all the elements of C. ( ) A =, B = 4 3. 3 3 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 Factory Bags 35 5 Suitcases 67 68

Chapter 5. Matrices, systems of linear equations and determinants Answer the following questions using matri operations. ) Knowing that the personnel costs represent /3 of the total cost, find the matri that gives the personnel cost of each product in each factory. ) Find the matri 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. euros for each product in each factory. 5.6 In this eercise we want to find a formula giving the successive powers of a diagonal matri. a) Compute A, A 3 and A 5 if A =. 3 b) Compute A 3. c) Let D be a diagonal n n matri whose diagonal entries are λ, λ,..., λ n. Make a guess about D r, for r Z, r, and prove it by induction. ( ) ( ) 5.7 Let A = and B =. Compute (AB) 3 t and B t A t. Observe that AB can be a non-symmetric matri even when both A and B are symmetric. 5.8 Give an eample of two matrices A and B such that (AB) t A t B t. 5.9 Let I be the identity matri and O the null matri of M (R). Find matrices A, B, C, D, E M (R) such that ) A = I and A I; ) B = O and B O; 3) C = C and C I, O; 4) DE = O but E D and ED O. 5. Let A and B be two symmetric matrices of the same type. Prove that AB is a symmetric matri if and only if A and B commute. 5. Do the following equalities hold for all matrices A, B M n (R)? If not, give a condition on A and B ensuring that the equations hold. ) (A + B) = A + B + AB; ) (A B)(A + B) = A B.

5.. Matri algebra 3 5. Let A and B be square matrices of the same type. A is said to be similar to B if there eists an invertible matri P such that B = P AP. If A is similar to B, prove: ) B is similar to A. In general we say that A and B are similar. ) The relation being similar is an equivalence relation. 3) A is invertible if and only if B is invertible. 4) A t is similar to B t. 5) If A n = O and C is an invertible matri of the same type as A, then (C AC) n = O. 5.3 Find a matri in row echelon form equivalent to each of the following matrices. Give the rank of the matri in each case. 5 6 3 4 3 3 ) 3 ) 3) 4 4) 3 6 4 3 8 4 3 5.4 Find the inverse of the following elementary matrices. ) ) ( ) ( ) 5 3) 4) 3 5) k, k 5.5 Using the Gauss-Jordan method, find, if it eists, the inverse of each of the following matrices. ) ( ) 3 5 ) 3 5 4 4 9 4) A = (a i,j ) 4 4, such that a i,j = if i j, and a i,j = otherwise. 5) A = (a i,j ) 4 4, such that a i,j = j if i j, and a i,j = otherwise. 3) 3 3 4 6) A = (a i,j ) 4 4, such that a i,i = k, a i,j = if i j =, and a i,j = otherwise.

4 Chapter 5. Matrices, systems of linear equations and determinants 5. Systems of linear equations 5.6 Which of the following equations are linear in, y and z? ) + 3y + z = ; ) y + + z = e ; 3) 4y + 3z / = ; 4) y = z sin π 4 y + 3; 5) z + y + 4 = ; 6) = z. 5.7 Find a system of linear equations for each of the following augmented matrices. 3 ) 3 4 5 ) 3) 4) ( 3 4 ) 5 /3 /4 /5 / 3 4 5.8 Answer the following questions. Justify your answers. ) What is the rank of the system matri of a determined consistent system with 5 equations and 4 unknowns? What about if the system is underdetermined? ) How many equations are needed at least to have an underdetermined consistent system with 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 unknowns? 4) Can a system with fewer equations than unknowns be inconsistent? 5) Give a determined consistent system, an underdetermined consistent system and an inconsistent system, each with 3 unknowns and 4 equations. 5.9 Solve the following systems of linear equations with coefficients in Z. Use Gaussian elimination and give the solution in parametric form. ) + y = + z = + y + z = ) + y = y + z = + z = 3) + y = y + z = + z =

5.. Systems of linear equations 5 5. Solve the following systems of linear equations. Use Gaussian elimination and give the solution in parametric form. ) ) + y + z = 8 y + 3z = 3 7y + 4z = 3y z = y + z = 3 y + 5z = 4 + y z = 3 y + z = 3) 4) y + z w = + y z w = + y 4z + w = 3 3w = 3 + 3 3 + 5 = + 6 5 3 4 + 4 5 3 6 = 5 3 + 4 + 5 6 = 5 + 6 + 8 4 + 4 5 + 8 6 = 6 5. Solve the following homogeneous systems of linear equations. Use Gaussian elimination and give the solution in parametric form. ) ) + y + z = + 5y + z = 7 + 7y + z = 4y + z + w = 5y + z = y z w = + 3y + w = y z + w = 3) 4) 3 3 + 4 = + 3 + 4 4 = + 3 8 4 = + 3 + 5 = + 3 3 4 + 5 = + + 3 + 5 = 3 + 4 + 5 = 5. Discus the following systems of linear equations according to the values of the parameters (assumed to be real). ) ) 3) + y + z = a + z = b + y + 3z = c b + y + z = b y + z = 3 y z = 6 y + z = 3b a + y z + t u = + ay + z t + u = + y + az + t u = y + z + at + u = + y z + t + au = 4) 5) + y 3z = 4 3 y + 5z = 4 + y + (a 4)z = a + + y = k a + by = k a + b y = k 3

6 Chapter 5. Matrices, systems of linear equations and determinants 5.3 Determinants 5.3 Assuming that ) ) d e f g h i a b c a b c d e f g h i a b c d e f g h i = 5, compute the following determinants. 3) 4) a + d b + e c + f d e f g h i a b c d 3a e 3b f 3c g h i 5.4 For which values of λ is the determinant of the following matrices equal to? ) ( λ ) λ 4 ) λ 6 λ 4 λ 4 5.5 Compute the following determinants. ) ) 3) 5 5 3 4 3 5 4 3 4) 5) 6) 4 6 8 8 6 4 3 4 3 4 3 3 7) 8) 3 8 8 6 4 6 9 7 4 8 4 6 4 3 6 8 5.6 Prove the following property. a a a 3 a a a 3 a 3 a 3 a 33 + b a a 3 b a a 3 b 3 a 3 a 33 = a + b a a 3 a + b a a 3 a 3 + b 3 a 3 a 33 5.7 For any matri A, let B be the matri obtained from A by adding to some row a multiple of another one. Prove that det(b) = det(a).

5.3. Determinants 7 5.8 Let A and B be 3 3 matrices such that det(a) = and det(b) =. Compute ) det(ab), ) det(a 4 ), 3) det(b), 4) det(a t ), 5) det(a ). 5.9 Prove that a a a a = (a + 3)(a ) 3.

6 Vector spaces A vector space over a field K consists of ) a non-empty set E, ) a binary operation E E E called addition and denoted +, and 3) a map K E E called scalar multiplication and denoted, satisfying the following eight properties for each u, v, w E and each λ, µ K: e) u + (v + w) = (u + v) + w (associative law); e) u + v = v + u (commutative law); e3) there eists a unique element E E such that u + E = u (zero vector); e4) for each u E there eists a unique u E such that u + u = E (additive inverse); e5) λ(µu) = (λµ)u; e6) λ(u + v) = λu + λv; e7) (λ + µ)u = λu + µu; e8) u = u, where denotes the multiplicative identity of K. (Note: in most cases, the field K will be R; other fields that may appear are Q and Z p.) Eercises 3 4 6 6. Let u =, v = and w = be vectors of R 3. Compute 8 4 ) u v; ) 5v + 3w; 3) 5(v + 3w); 4) (w u) 3(v + u).

9 6. Draw the following vectors of R. ) v = ( ) ; ) v 6 = ( ) 4 ; 3) v 8 3 = ( ) ; 4) v 5 4 = ( ) 3. 6.3 For the vectors in the previous eercise, compute graphically the vectors v + v, v v 3 and v v 4 and check your answers algebraically. 6.4 Let u, v, w be elements of a vector space and let α, β, γ be elements of the corresponding field of scalars, with α. Assuming that the relation αu + βv + γw = holds, write the vectors u, u v and u + α βv in terms of v and w. 6.5 Let P (R) e be the set of all polynomials with coefficients in R and with only even powers of. Determine if P (R) e is a vector space with the usual addition and scalar multiplication of polynomials. (Assume that the polynomial has degree.) 6.6 Let F(R) be the set of all functions f : R R. Given two functions f, g F(R) and λ R, let us define the functions f + g and λf by (f + g)() = f() + g(), (λf)() = λf(). Prove that F(R) is a R-vector space with these operations. 6.7 Which of the following sets are vector subspaces over R? (Justify your answers.) ( ) E = { R y : + πy = }, E 5 = { y z R4 : + y + z + t =, t = }, t ( ) E = { y R 3 : + z = π}, E 6 = { R y : + y + y = }, z a + b E 3 = { y R 3 : y = }, E 7 = { a b c R4 : a, b, c R}, z a + c ( ) a E 4 = { R y : Q}, E 8 = { a b + a R4 : a, b R}. + a

Chapter 6. Vector spaces 6.8 Let P (R) be the vector space of all polynomials in with coefficients in R. Which of the following subsets are vector subspaces of P (R)? (Justify your answers.) F = {a 3 3 + a + a + a P (R) : a = a } F = {p() P (R) : p() has degree 3} F 3 = {p() P (R) : p() has even degree} F 4 = {p() P (R) : p() = } F 5 = {p() P (R) : p() = } F 6 = {p() P (R) : p (5) = } 6.9 Let M n m (R) be the vector space of all n m matrices with coefficients in R. Which of the following subsets are vector subspaces of M n m (R)? (Justify your answers.) {( ) ( ) ( ) ( ) ( )} a b a b a b M = M c d (R) : = c d c d M = {A M n m (R) : A = A t } M 3 = {A M n m (R) : a i = i [m]} M 4 = {A M n m (R) : a i = i [m]} M 5 = {A M n m (R) : AB = } (where B is a fied matri) 6. Let T R 4. Prove that the vector u below can be written as a linear combination of the vectors in T in at least two different ways. T = {,,, }, u = 3 5 6. For which values of the parameter a can the vector u R 3 be written as a linear combination of the vectors in T? 3 u = 5, T = {, 7, } a 6. Give the values of the parameters a and b for which the matri ( ) ( ) 4 combination of and. 5 3 ( ) a b is a linear 6.3 Given the vectors u = and v = of R 3, find a condition on the components of a vector y which ensures that this vector is in the subspace spanned by {u, v}. z

6.4 Give the form of a generic polynomial in P (R) that belongs to the vector subspace spanned by the set { +, }. 6.5 Let F =, and G =, be subspaces of R 3. ) Prove that F = G. 9 ) Let e =. Prove that e F and epress it as a linear combination of the two families of vectors that span F. 6.6 Determine whether the following sets of vectors are linearly independent in the respective vector spaces. 3 ) {, 6 } in R 3 ; 3 8 3 ) { 3,, 4 } in R 3 ; 5 3 5 3 8 3) { 4, 3, } in R 3 ; 3 3 4 6 4 4) { 5, 3 3, 5 } 6 3 9 6 R 4 ; 5) { 3 4 3 4 } in R5. 5 4 7 in 6.7 Determine whether the following vectors in three-dimensional space are coplanar, collinear, or neither. 6 ) {,, }; 4 4 6 3 4 ) { 7,, }; 4 3 3) {, 4, 6 }; 3 6 4 4) { 6, 3, 3 }. 8 4 4 3 3 6.8 Let us consider the vectors, b and 5 a in the vector space R4. Determine a 4 a and b so that they are linearly dependent vectors, and in that case epress the vector R 4 as a non-trivial linear combination of them.

Chapter 6. Vector spaces 6.9 Let E be a R-vector space and let u, v, w be any three vectors in E. Prove that the set {u v, v w, w u} is linearly dependent. 6. Prove that the following matrices A, B and C are linearly independent as vectors in M 3 (R). A = ( ) (, B = ) (, C = 3 ). Prove that for any λ the matri is a linear combination of A and B. ( λ 4 λ ) 6. Prove that the polynomials +, + and + are linearly dependent in P (R). 6. If {e, e,..., e r } is a family of linearly dependent vectors in some vector space, is it true that any vector e i can be written as a linear combination of the other vectors? Prove it or give a countereample. 6.3 Determine whether the following statements about a family of vectors in a vector space E are true, proving them when they are true and giving a countereample otherwise. ) If {e,..., e r } are linearly independent and v e i for each i, then the set {e,..., e r, v} is linearly independent. ) If {e,..., e r } are linearly independent and v e,..., e r, then {e,..., e r, v} is linearly independent. 3) If {e,..., e r } spans E and v e i for each i, then {e,..., e r, v} also spans E. 4) If {e,..., e r } spans E and e r e,..., e r, then {e,..., e r } also spans E. 5) Every set with only one vector is linearly independent. 6.4 Let us consider the set of vectors {,,, }. 4 ) Prove that they are a basis of R 4. ) Give the coordinates of the vector 3 in this basis.

3 3) Give the coordinates of an arbitrary vector y z in this basis. t {( ) ( ) ( ) ( )} 6.5 Let B =,,,. Prove that B is a basis of M (R). ( ) 3 Give the coordinates of A = in this basis. 3 6.6 Let P 3 (R) be the vector space of all polynomials of degree at most 3. Prove that the polynomials +, +, + and + 3 form a basis of P 3 (R) and give the coordinates of the polynomial 3 + 3 + 6 5 in this basis. 4 6.7 Let F =,, in R 3. Give a basis of F and find the condition that the components of y must satisfy for this vector to belong to F. z 6.8 Consider the following subspaces of R 4. F =,,, G =,, 3 Prove that F = G and that the respective spanning sets are bases. Do the vectors and belong to F? If so, give their coordinates in the two bases. 6.9 Let {v, v, v 3 } be a basis of a vector space E. Prove that the set {v + v, v + 3v 3, 3v 3 + v } is also a basis of E.. 6.3 Give a basis of the subspace E of R 5 and complete it to a basis of R 5 for E = { 3 4 R5 : 3 = + 4, 5 = }. 5

4 Chapter 6. Vector spaces 6.3 Let us consider the vectors in M (R) ( ) 4, ( 6 ), ( ). Prove that they are linearly independent and give a vector that together with them forms a basis of M (R). 6.3 For which values of λ the vectors dimension? λ λ,, λ span a vector subspace of R4 of λ λ 6.33 In each case, give a basis and the dimension of the spaces E, F and E F : ) E = { y R 3 : = y = z} and F = { y R 3 : + y = z, 3 + y + z = } as z z subspaces of R 3. 4 ) E =,, and F =, 3, 3 as subspaces of R 3. a a 3) E = { a + 3b a b : a, b, c R} and F = { b : a, b R} as subspaces of R4. c 3b {( ) } ( ) ( ) a b 4) E = M c d (R) : a = b = c and F =, as subspaces of M (R). 6.34 For each of the subspaces E in the previous eercise (eercise 6.33), complete the respective bases to a basis of the whole vector space. 6.35 Let M (R) be the space of all real matrices and let S and T be the subsets S = {A M (R) : A = A t } and T = {A M (R) : A = A t }. ) Prove that S and T are vector subspaces. ) Give a basis and the dimension of each subspace. 3) Prove that every matri M M (R) can be written as M = A + B, with A S and B T. 6.36 Give two vector subspaces of R 4 of dimension two such that their intersection reduces to a single vector. Is this possible in R 3? Justify the answer.

5 6.37 Let B = { 5, 5, 4 } and B = { 3,, } be bases of R 3. 6 3 5 4 ) Prove that they are indeed bases. ) Give the change-of-basis matri from B to B (PB B ) and the change-of-basis matri from B to B (PB B ). 3) Compute the coordinates in both bases B and B of the vector that in the canonical basis has coordinates 5. 6.38 Let B = {p (), p (), p 3 ()} be a basis of P (R), the space of all polynomials of degree. Let u() = + +, v() = + 3 and w() = +. If the coordinates of u(), v() and w() in the basis B are u() B = (,, ), v() B = (,, ) and w() B = (,, ), respectively, give the coordinates of the vectors in B in the canonical basis {,, }. 6.39 Let B = B = {( ) ( ) ( ) ( )},,,, {( ) ( ) ( ) ( )},,, be two bases of M (R). Give the change-of-basis matrices PB B B and PB. 6.4 Given B and B below, prove that they are basis of R 3. B = {,, }, B = {,, }. Let u be a vector of R 3 whose coordinates in the basis B and B are u B = y and u B = z y, respectively. Epress, y and z in terms of, y and z, and conversely. z 6.4 Let E be an R vector space of dimension and let B = {e, f} be a basis of E. Define u = 3e f and v = e + f. ) Prove that B = {u, v} is also a basis of E. ) Give the coordinates of u, v, e and f in the bases B and B. ( ) 3 3) Let w be the vector whose coordinates in the basis B are. Give its coordinates in 4 the basis B.

7 Linear maps Eercises 7. Determine which of the following maps are linear: ( ) ( ) ) f : R R, f = + y; y 4) f : R R 3, f = y ; y ( ) + y ) f : R R, f = y y ; + 7 y 3) f : R 3 R 3, f y = y ; 5) f : R 3 R 3, f y = z. z + y + z z 7. Determine which of the following maps f : P (R) P (R) are linear: ) f(a + a + a ) = ; ) f(a + a + a ) = a + (a + a ) + (a 3a ) ; 3) f(a + a + a ) = a + a ( + ) + a ( + ). 7.3 Determine which of the following maps are linear: ( ) a b ) f : M (R) R defined by f( ) = a + d; c d ) f : M (R) M 3 (R) defined by f(a) = AB, with B M 3 (R) a fied matri; 3) f : M n (Z ) Z, defined by f(a) = det(a). 7.4 In each case, determine if there eists an endomorphism f : R 3 R 3 such that f(u i ) = v i, i =,, 3,: 3 8 ) u =, u =, u 3 = and v =, v =, v 3 = 4 ; 3

7 3 8 ) u =, u =, u 3 = and v =, v =, v 3 = 4 ; 3 3 3 3) u =, u =, u 3 = and v =, v =, v 3 =. 3 3 7.5 Let f : P (R) P (R) be the linear map defined by: f() = +, f() = 3 and f( ) = 4 + 3. What is the image of the polynomial a + a + a? Compute f( + 3 ). 7.6 Let f : E F be a linear map. ) Prove that the preimage of the zero vector of F is a subspace of E. It is called the kernel of f and it is denoted Kerf. ) Is the preimage of any non-zero vector of F a subspace? 7.7 Determine whether the following linear maps are bijective or not using the information given: ) f : R n R n, with Kerf = { R n}; ) f : R n R n, with dim (Im f) = n ; 3) f : R m R n, with n < m; 4) f : R n R n, with Im f = R n. 7.8 Let E and F be two vector spaces of dimensions n and m, respectively. Let f : E F be a linear map. Prove that ) if f is injective, then n m; ) if f is surjective, then n m; 3) if n m, then f is not an isomorphism. 7.9 For each of the following linear maps, give the associated matri in the canonical bases; give the dimension and a basis of the kernel and of the image of the map; determine if the map is injective, surjective, bijective or none of them; and find the inverse map, if it eists: ) f : R R, f() = a, for given a R;s ( ) ( ) + 3y ) f : R R, f = ; y + 7y y + z + t 3) f : R 4 R 3, f y z = t y z + t y + z ;

8 Chapter 7. Linear maps 4) f : P (R) P (R), f(a + a + a ) = (a a ) + (a a ) + (a a ) ; 5) f : P (R) P (R), f(a + a + a ) = 3a + (a a ) + (a + a + a ) ; ( ) 6) f : Z Z 3, f = y ; y y + y + z 7) f : Z 3 Z 3, f y = z y + z z ; ( ) ( ) a b a + d 8) f : M (R) R, f( ) = ; c d b + c ( ) 9) f : R 3 M (R), f y y y z =. z y z z y 7. Find the kernel of the linear map f : R 3 R 3 given by f y = y z, compute z z f and the preimages of the vectors and. 7. Let B be an invertible matri n n. Prove that the map f : M n (R) M n (R) defined by f(a) = AB is a bijective endomorphism. 7. For the following linear maps f and f, determine whether the composition function f = f f is injective, surjective or bijective. z ) f : R 3 R 3 and f : R 3 R, where f y = + y and f y = z + z y z ( ) 3z ; y + 4z ) f : P 3 (R) P (R) and f : P (R) P (R), where f (a + a + a + a 3 3 ) = a + a 3 + a and f (a + a + a ) = (a + a ) + (a + a ) + (a + a ) ; ( ) y + z 3) f : Z Z 3 and f : Z 3 Z 3, where f = + y and f y y = + y + z. y z y + 7.3 Let E be an R-vector space and B = {u, v, w, t} a basis of E. Let f be an endomorphism of E such that f(u) = u + w, f(v) = v + w, f(w) = u + v + w, f(t) = u + v + 4w. Find the matri of f in the basis B, and find a basis and the dimension of the image of f.

9 7.4 Let f : E F be a linear map such that f(e e ) = u, f(e e 3 ) = u u and f(e 3 ) = u + u 3, where B = {e, e, e 3 } is a basis of E and B = {u, u, u 3, u 4 } is a basis of F. Give the dimension of the image of f. 7.5 For the following subspaces E and F of R 4, determine if there eists a linear map f : R 4 R 4 such that f(u) = for each u E and f(v) = v for each v F. ) E = 3 4 ) E =. 3 5 7.6 Let f be an endomorphism of R 3 with associated matri (m ) m. m (m + ) m + Find the dimension of the image according to the values of m. 7.7 Let E and F be two K-vector spaces, f : E F a linear map, and v, v,..., v n vectors of E. For each of the following statements, prove it if it is true and give a countereample if it is false. ) If v, v,..., v n are linearly independent, then f(v ), f(v ),..., f(v n ) are linearly independent. ) If f(v ), f(v ),..., f(v n ) are linearly independent, then v, v,..., v n are linearly independent. 3) If v, v,..., v n is a spanning set of E, then f(v ), f(v ),..., f(v n ) is a spanning set of F. 4) If f(v ), f(v ),..., f(v n ) a spanning set of F, then v, v,..., v n is a spanning set of E. 5) If v, v,..., v n is a spanning set of E, then f(v ), f(v ),..., f(v n ) is a spanning set of Im f. 7.8 Give the matrices of the following linear maps in the canonical basis: ) f : R R such that f ) f : R R 4 such that f ( ) = ( ) = ( ) and f 3 ( ) ( = ) ; ( ) 4 and f = 3 ;

Chapter 7. Linear maps 3) f : Z 3 Z such that f = ( ), f = ( ) and f = ( ). 7.9 Let f be the endomorphism of P (R) given by f(a + a + a ) = 3a + (a a ) + (a + a + a ). Give the matri of f in the basis B = { +, + +, + + }. 7. Let B E = {e, e, e 3 } be a basis of an R-vector space E and B F = {v, v } a basis of an R-vector space F. Let us consider the linear map f : E F defined by Find the matri of f in the indicated bases: f(e + ye + ze 3 ) = ( z)v + (y + z)v. ) B E = {e, e, e 3 } and B F = {v, v }. ) B E = {e, e, e 3 } and B F = {v, v }. 3) B E = {e + e + e 3, e + e 3, 3e 3 } and B F = {v, v }. 4) B E = {e + e + e 3, e + e 3, 3e 3 } and B F = {v, v }. 7. Let us consider the endomorphism f N : M (R) M (R) defined by f N (A) = NA where ( ) N =. ) Find the matri of f N in the canonical basis of M (R). ) Compute Kerf N and Im f N. 3) Find the matri of f N in the basis B = {( ), ( ), ( ), ( )}. 7. Let M = 3 be the matri of an endomorphism f of R 3 in canonical basis. ) Find the subspaces Kerf and Imf. ) Find a basis B of R 3 such that the matri of f in basis B is M B =.

7.3 Let u = and u = be vectors of R4 and let E be the subspace spanned by B E = {u, u }. Let v = and v = be vectors of R 3 and F the subspace spanned by B F = {v, v }. Let f : E F be given by f( u + u ) = ( )v + ( + )v. ) Find the matri of f in the bases B E and B F. ) Is f injective? Is it surjective? 3) Let B E = {, } and B F = {, }. Prove that they are bases of E and F, respectively, and give the matri of f in these new bases. 7.4 Using the matri of the corresponding linear map, give the image of the vector by the reflection through ) the ais; ) the y ais. ( ) 7.5 Using the matri of the corresponding linear map, give the image of the vector 5 3 by the reflection through ) the plane y; ) the plane yz; 3) the plane z. 7.6 ( Using ) the matri of the corresponding linear map, give the orthogonal projection of the vector onto 5 ) the ais; ) the y ais. 7.7 Using the matri of the corresponding linear map, give the orthogonal projection of the vector 5 onto 3 ) the plane y; ) the plane yz; 3) the plane z.

Chapter 7. Linear maps 7.8 Find the image of the vector ( ) of R 3 by a rotation of angle ) θ = 3 o ; ) θ = 6 o ; 3) θ = 45 o ; 4) θ = 9 o. 7.9 Find the image of the vector by a counterclockwise rotation, ) of 3 o around the ais; ) of 45 o around the y ais; 3) of 9 o around the z ais; 4) of 6 o around the z ais. 7.3 Give the matri of the following compositions of linear maps of R : ) a counterclockwise rotation of 3 o, followed by a reflection through the y ais; ) an orthogonal projection onto the y ais, followed by a contraction of factor k = /; 3) a dilation of factor k =, followed by a counterclockwise rotation of 45 o, followed by a reflection through the y ais. 7.3 Give the matri of the following compositions of linear maps of R 3 : ) a reflection through the yz plane, followed by an orthogonal projection on the z plane; ) a counterclockwise rotation of 45 o around the y ais, followed by a dilation of factor k = ; 3) a counterclockwise rotation of 3 o around the ais, followed by a counterclockwise rotation of 3 o around the z ais, followed by a contraction of factor k = /3. 7.3 Let f : R R and f : R R be linear maps. Determine if f f = f f when: ) f and f are the orthogonal projections onto the y and aes, respectively; ) f is the counterclockwise rotation of angle θ and f is the counterclockwise rotation of angle θ ; 3) f is the reflection through the ais and f is the reflection through y ais; 4) f is the orthogonal projection on the y ais and f is the counterclockwise rotation of angle θ. 7.33 Let f : R 3 R 3 and f : R 3 R 3 be linear maps. Determine if f f = f f when: ) f is a dilation of factor k and f is a counterclockwise rotation around the z ais of angle θ; ) f is a counterclockwise rotation around the ais of angle θ and f is counterclockwise rotation around the z ais of angle θ.

8 Diagonalization Eercises (In this chapter, all vector spaces are of finite dimension.) 8. Let f be an endomorphism of a K-vector space E and let u E be an eigenvector of f of eigenvalue λ K. Prove that: ) u is an eigenvector of f of eigenvalue λ; ) u is an eigenvector of f of eigenvalue λ. 8. Prove that a square matri A and its transpose A t have the same eigenvalues. 8.3 Let J be the matri of M 5 (R) with all entries equal to. Find a basis of R 5 consisting of an eigenvector of J of eigenvalue 5 and 4 eigenvectors of eigenvalue. 8.4 Let E be an R-vector space and f and g two endomorphisms of E with the same eigenvalues. Prove: ) f is bijective if and only if is not an eigenvalue of f; ) f is bijective if and only if g is bijective. 8.5 Let A M n (R). Suppose tht there eist m N, m >, and R n such that A m R n and A m = R n. Show that A m is an eigenvector of A. 8.6 Compute the characteristic polynomial, the eigenvalues and the eigenspaces of the following matrices. In each case, determine if they are diagonalizable and give, when it eists, a basis in which the associated matri is diagonal. ) ( ), ) 3 ( ), 3) 3 4, 3

4 Chapter 8. Diagonalization 4) 5), 3 3 3 5 3, 6 6 4 6) 7) 4 4,, 8), 4 3 6 9) 4 3 3. 4 8.7 Find the eigenvalues and eigenvectors of the following endomorphisms. If they are diagonalizable, give a basis in which they diagonalize and the associated diagonal matri. + 4z ) f : R 3 R 3, f y = 3 4y + z. z y + 5z ) f : P (R) P (R), f(a + b + c ) = (5a + 6b + c) (b + 8c) + (a c). 3) f : P 3 (R) P 3 (R), f(a + b + c + d 3 ) = (a + b + c + d) + (b + c + d) + 3(c + d) + 4d 3. 8.8 Find the eigenvalues and eigenvectors of the endomorphism f : M (R) M (R) defined by ( ) ( ) a b c a + c f =. c d b c d 8.9 Prove that if A M n (R) is an upper triangular matri, with the elements in the diagonal pairwise different, then A is diagonalizable. 8. Discuss whether the following matrices diagonalize over R according to the values of the parameters: ) ) 3) ( a ) b b a, b c a b, 4) 5) c a b a, b c a, b 6) 7) a a a a a, a a b a.

5 8. Let f : R 3 R 3 be a linear map. Let F and G be the vector subspaces of R 3 defined as follows: F = { y R 3 + z = } and G =,. z Consider the following three conditions on f, F and G: (I) f = and f f = f ; (II) f = f; (III) f(f G) F G and f(f G) { }. Prove that: ) if (I) holds, then and are eigenvalues of f; ) if (II) holds, then the only possible eigenvalues of f are and ; 3) F G = 3 ; 4) if (III) holds, then 3 is an eigenvector of nonzero eigenvalue; 5) if all three conditions (I), (II) and (III) hold, then f is diagonalizable; find a basis in which f diagonalizes. 8. Determine if there eists an endomorphism f : R 3 R 3 satisfying the conditions specified below. If it eists, give it eplicitly, compute its characteristic polynomial and determine if it is diagonalizable or not. ) and are eigenvectors of eigenvalue and is an eigenvector of eigenvalue. ) f () = { y R 3 5 + y z = } and is an eigenvector of eigenvalue /. z 3) f =, f = and f =.

6 Chapter 8. Diagonalization 4) f () = and F = { y R 3 + y = z} is the eigenspace of the eigenvalue. z a y 8.3 Consider the endomorphism f of R 3 defined by f y = + y + z, where a is a z z real parameter. ) Give the dimension of Im f according to the values of a R. ) Let F and G be the subspaces of R 3 defined by F = { y R 3 ay + z = } and G = { y R 3 b + y + bz = }, z z where b is another real parameter. Find the values of a and b such that G is equal to the image of F by f. 3) Is f diagonalizable when a = 3? 4) Give conditions on a such that all eigenvalues of f are real. 8.4 Let A M n (R). ) How are the eigenvalues of A and A k related? And the eigenvectors? ) Prove that if the matri A can be written as A = P DP, where P is an invertible matri, then A k = P D k P. 3) Using the previous result, compute i) ( ) 3 7 6, ii) 35 3 9, iii) 5 3 5 7 7 9 7. 8.5 An UFO leaves a planet in which the vectors v, v, v 3 have their origin. These vectors are used as a basis of a coordinate system of the universe (R 3 ). After arriving at the point the spaceship is pushed by a strange force such that every day it is moved from the point v to the point Av, where A =. 3 ) What will be the position of the spaceship after days? ) Will the spaceship arrive some day to Earth, which is located in the point ( ) 3? 5

Review eercises Matrices, systems of linear equations and determinants B. Given two diagonal matrices A and B of the same type, prove that AB = BA. B. Let A and B be two upper (lower) triangular matrices of the same type. Prove that AB is an upper (lower) triangular matri. B.3 Prove that if A is an m n matri, then AA t and A t A are symmetric matrices. B.4 Let A be a symmetric n n matri such that A = A, and let us assume that for any i we have a ii =. Prove that all entries in the ith row and in the ith column of A are zero. B.5 Find the set of matrices A M (R) such that A = I. B.6 Let A, B and C be matrices. Prove that ) if A is row equivalent to B, then B is row equivalent to A; ) if A is row equivalent to B and B is row equivalent to C, then A is row-equivalent to C. B.7 Prove that the following system of linear equations is consistent but underdetermined with two degrees of freedom. Answer the following questions. 4 y + z + t + u = y + z u = + z + t = y + t + u = 5 + y + 3z + t u = 3 ) What is the maimum number of independent equations? ) Is the second equation a linear combination of the others? Answer the same question for the fourth equation.

8 Review eercises 3) Is there any solution of the system such that = π? Answer the same question for y = 3. 4) Is there any solution of the system such that = and y = 3? The same for z = π and t = 3. B.8 For any matri A let B be the matri obtained by multiplying one row of A by a constant c. Prove that detb = c deta. B.9 Solve the following systems of linear equations. Use Gausssian elimination and give the solution in parametric form. ) ) + y = + 3y = + 4y = + 3y = 3 + y + z = + 3y + z = 5 + 4y + z = 4 3) 4) y + z = 7 y + z 4w = + y z + w = 4 3 + y 8z w = 9 + 6y z 4w = y + 5z + 7w = 3 + 8y z 6w = 5 + 3y 6z 3w = B. Discuss the following systems according to the values of the parameters (assumed to be real). ) ) + y + mz = m + my + z = m m + y + z = m { (a ) ay = 6a (a )y = 5 a 3) 4) a + 3y = 3 + y = a + ay = 3 a + y + 3z + u = 6 + 3y z + u = b 3 ay + z = 5 + 4y + 3z + 3u = 9 Vector spaces B. Let F(R) be the vector space defined in the eercise 6.6. Determine which of the following subsets are vector subspaces of F(R). (Justify your answers.) F = {f F(R) : f() = f()} F = {f F(R) : f() = f() + } F 3 = {f F(R) : f() = } F 4 = {f F(R) : f( ) = f() R} F 5 = {f F(R) : f is continuous}

Review eercises 9 B. Prove that the set of solutions of a system of linear equations with n unknowns is a vector subspace of R n if and only if the system is homogeneous. B.3 Let us consider the subspaces of R 3 F =,, 3 and G =, 4. 3 a Determine the values of a such that F = G. B.4 Let E be an R-vector space and u, v, w three vectors such that u+v w = 4u 5v+w. Prove that {u, v, w} is a linearly dependent set. B.5 Let E be a vector space and v, v, v 3 vectors of E. Prove that the following statements are equivalent. a) {v, v, v 3 } is linearly independent. b) {v + v, v, v + v 3 } is linearly independent. c) {v + v, v + v 3, v + v 3 } is linearly independent. B.6 Prove that the set {,,, } is a basis of R4. Write down the vector 3 7 6 as a linear combination of the vectors in this basis. 5 B.7 Let F be the subspace of R 4 given by ) Find a basis of F. F = 5,, 3, 3 4, 9 4 5 9 ) Prove that e = 7 F and compute the coordinates of e in the basis of the previous 79 question. 3) Find a basis of F that contains e. 4 5 37.

3 Review eercises B.8 Let {u, v} be a basis of R. Prove that the set {αu+βv : α+β = } is a vector subspace. Describe this subspace geometrically and find a basis of it. B.9 Let E be a vector subspace. Prove: ) If {e,..., e n } is a spanning set of E such that when one removes any of the e i it no longer spans E, then {e,..., e n } is a basis of E. ) If {e,..., e n } is a linearly independent set of E such that it becomes linearly dependent after adding any new vector, then {e,..., e n } is a basis of E. B. Find a basis and the dimension of the following subspaces. ) E = { y z R4 : = y = z}. t ) E = { y z R4 : = y 3z, z = t}. t 3) E 3 =,,, (Z 5 ) 3. 4) E 4 =, 3, (Z 5 ) 3. 4 5) E E. 6) E 3 E 4. B. Let E, E, E 3 and E 4 be the subspaces in the previous eercise. For each of them, complete the basis found to a basis of the whole space. B. Let F = { a a : a, b Q}. Prove that it is a vector subspace of Q 3. Find a basis b a and its dimension. Answer the same questions if we take Z instead of Q as the field of scalars. B.3 A matri M M n (R) is called magic if the sum of the elements in each row, column and in the main diagonal is the same in all cases. Prove that the set of magic matrices is a subspace of M n (R). For n =, 3 find a basis of this subspace and its dimension.

Review eercises 3 3 3 6 B.4 Let B = {,, 6 } and B = { 6, 6, 3 } be bases of R 3. 3 4 7 ) Prove that they are indeed bases. ) Give the change-of-basis matri from the basis B to the basis B (PB B ) and the changeof-basis matri from B to B (PB B ). 3) Compute the coordinates in the bases B and B of the vector v whose coordinates in the 5 canonical basis are 8. 5 B.5 Let E be a vector space over R of dimension 3 and let {e, e, e 3 } be a basis of E. The vector v has coordinates in this basis. Compute the coordinates of v in the basis: (You do not need to prove that it is a basis.) u = e + e + e 3 u = e e u 3 = e e 3 B.6 Let B = B = {( ) ( ) ( ) ( )},,,, {( ) ( ) ( ) ( )} 3 5 3,,, 5 4 4 5 be two bases of M (R). Give the change-of-basis matrices PB B B and PB. Linear maps B.7 Let B = {v, v, v 3 } be a basis of R 3, with v =, v = 5 and v 3 =. Is 3 3 ( ) ( ) ( ) there a linear map f : R 3 R such that f(v ) =, f(v ) =, and f(v 3 ) =? If so, give the image of the vector y by this map f. z B.8 Determine which of the following vector spaces are isomorphic to R 6 : M 3 (R), R 6 [], M 6 (R), W = {(,, 3,, 5, 6, 7 ) : i R}.

3 Review eercises B.9 Let f : M 3 (R) M 3 (R) be defined by f(a) = A A t. Compute the preimage of the zero vector and its dimension. B.3 Let B = {u, v, w} be a basis of a K-vector space E, and let f be an endomorphism of E such that: f(u) = u + v, f(w) = u, Kerf = u + v. Give a basis and the dimension of the subspaces Imf and Imf B.3 Let f be an endomorphism of R 3 such that f = and f () = { y z R 3 : 3y + z = }. ) Give the matri of f in the canonical basis. ) Compute the dimension of f () and of the image of f. 3) Give an eplicit epression for the image of an arbitrary vector. 4) Determine if it is injective, surjective, bijective or none of them. B.3 Let f r be the endomorphisms of R 3 defined by + y + z f r y = z + y + y + rz, r R. ) Find the value of r such that the dimension of Im f r is the lowest possible one. ) For the values of r obtained in the previous item, give a basis and the dimension of the preimage of the zero vector by f r. 3 3) Given v =, does it eist any t such that fr () = v? t 4) If w =, compute f r (w) and fr (f r (w)). B.33 Let E and F be two K-vector spaces and f : E F a linear map. Let v, v be two vectors of E. Prove the following assertions: ) If v and v are linearly dependent, then f(v ) and f(v ) are linearly dependent. ) If f(v ) and f(v ) are linearly independent, then v and v are linearly independent. 3) Assuming f is injective, if f(v ) and f(v ) are linearly dependent, then v and v are linearly dependent.

Review eercises 33 B.34 Let E be a R-vector space and v, v,..., v n vectors of E. Let us also consider the map f : R n E given by f(,,..., n ) t = v + v + + n v n. Prove the following assertions. ) f is injective if and only if the vectors v, v,..., v n are linearly independent. ) f is surjective, if and only if, the vectors v, v,..., v n span E. 3) f is bijective, if and only if, the vectors v, v,..., v n are a basis of E. ( ) B.35 Let A = be the matri of f : Z 3 Z in the bases B = {,, } ( ) ( ) of Z 3 and B = {, } of Z. ) Give the matri of f in the canonical bases of Z 3 and Z. ( ) ( ) ) Find the matri of f in the basis B = {,, } of Z 3 and B = {, } of Z. 3) Let v Z 3 be the coordinates in the basis B. Give the coordinates of f(v) in the basis B. B.36 Let f be an endomorphism of R 3 whose matri in the basis u =, u =, u 3 = is given by 5 3 A = 5 3. 8 5 ) Find the matri of f in the canonical basis. ) Determine if f is injective, surjective or bijective. 3) Find the preimage of the vector w = au + bu bu 3 according to the values of a and b. Give the result in the canonical basis. B.37 Let B = {e, e, e 3, e 4 } be a basis of a Z -vector space E, and let f : E E be the linear map defined by f( e + e + 3 e 3 + 4 e 4 ) = ( + )e + ( 3 + 4 )e.

34 Review eercises ) Find the matri of f in the basis B. ) Prove that f f and f = f 3. 3) Find a basis and the dimension of the subspaces Imf, Imf. 4) Prove that the vectors e 4, e + e, e + e 4, e + e + e 3 form a basis of E and write the matri of f in this basis. B.38 Give the matri of the following compositions of the linear maps of R : ) a reflection through the ais, followed by a dilation of factor k = 3; ) a counterclockwise rotation of 6 o, followed by an orthogonal projection onto the ais, followed by a central symmetry. B.39 Give the matri of the following compositions of linear maps of R 3 : ) an orthogonal projection onto the y plane, followed by a reflection through the yz plane, followed by a contraction of factor k = 3; ) a reflection through the y plane, followed by a reflection through the z plane, followed by a counterclockwise rotation of 6 o around the z ais. Diagonalization B.4 Let f : E E be a bijective endomorphism. Prove that f diagonalizes in the basis B if and only if f diagonalizes in the basis B. What is the relationship between the eigenvalues of f and those of f? B.4 Let A M n (R) be a matri such that A = A, and let B M n (R) be any matri. Prove that any eigenvalue of AB is also an eigenvalue of ABA. B.4 For each of the following endomorphisms, find the characteristic polynomial, the eigenvalues and the eigenspaces. Determine if they are diagonalizable. ( ) ( ) ) f : R R, f =. y + 3y ) f : R 3 R 3, f y = 3y. z z z 3) f : R 3 R 3, f y = 3y. z 4) f : R 3 R 3, f y = 8 y + 36z. z + 4z

Review eercises 35 + z 5) f : R 3 R 3, f y = 3 + y + 3z. z + z 6) f : R 4 R 4, f y z + 5y + 6z + 7t 3 + 8z + 9t. t 4 + t 7) f : R n R n, f... = n + + + n + + n... n. B.43 Let us consider the endomorphism f of R 4 whose matri in canonical basis is A =. ) Compute the characteristic polynomial and the eigenvalues. ) Find a basis of R 4 given by eigenvectors of f. 3) Give the change-of-basis matri P from the basis obtained in ) to the canonical basis. Compute P. 4) Write down the diagonal matri of f in the basis obtained in ) and prove that it is equal to P AP. B.44 Let C M n (R) be a non-null matri and let f C : M n (R) M n (R) be the map defined by f C (A) = CA AC, for each matri A M n (R). ) Prove that f C is not injective. ( ) a ) If C =, for which values of the real parameters a, c and d the endomorphism f c d C diagonalizes?