MATH 218 Final, Fall 2008

Fall 2008-2009 Final Exam Date: January 29 Prof: M. Egeileh Math 218 - Section 7 Duration: 2 hours Exercise 1 The vector space M2(R) being equipped with its standard inner product (defined by < A, B >= tr((tA)B) for any A, B ∈ M2(R)), we consider the following two matrices : A = � 8 0 0 0 � and B = � 3 √ 3 0 0 3 � . 1. Compute ∥A∥ and ∥B∥. (10 pts) 2. Find the angle between the matrices A and B. (10 pts) Exercise 2 In this exercise, B = {e1, e2, e3} is the standard basis of R3 and B′ = {e′ 1, e′ 2} is the standard basis of R2. We consider the linear map T : R3 −→ R2 defined by : T(x1, x2, x3) = (x1 − x2 + 3x3 , 4x1 − 4x2 + 12x3) for every (x1, x2, x3) ∈ R3. 1. Let A = [T]B′,B be the representative matrix of T in the bases B and B′. Write A (find all its entries). (5 pts) 2. Check that T(x) = Ax for every x ∈ R3 (here, x and T(x) are written as column vectors). (3 pts) 3. Find a basis for the kernel of T. (6 pts) 4. What is the rank of T ? Justify your answer. (4 pts) 5. Find a basis for the image of T. (6 pts) 6. Is T injective ? surjective ? bijective ? Justify your answers. (6 pts) Exercise 3 Let V be a 3-dimensional vector space, and B = {a1, a2, a3} a basis of V . We consider the linear operator T : V −→ V whose representative matrix in B is the following matrix : A =   1 −1 −2 0 0 0 0 0 −1  . The two parts of the problem can be treated independently ; however, it is recommended to solve Part I before Part II. Part I : 1. Find the eigenvalues λ1, λ2, λ3 of T. (6 pts) 2. Without performing any computation, find, for each eigenvalue λi you obtained in 1., the dimension of the corresponding eigenspace Ker(T − λiIdV ) (Hint : remember that for any eigenvalue λ, dim Ker(T −λIdV ) (the geometric multiplicity of λ) is not greater than the multiplicity of λ as root of the characteristic polynomial of A). (5 pts) 3. Deduce that the linear operator T is diagonalizable. (4 pts) 4. Find, for each eigenvalue λi, a basis of the corresponding eigenspace Ker(T−λiIdV ). (6 pts) Part II : 1. Let a′ 1, a′ 2, a′ 3 be the vectors of V defined by : a′ 1 = a1, a′ 2 = a1 + a2, a′ 3 = a1 + a3. Check that B′ = {a′ 1, a′ 2, a′ 3} is a basis of V . (2 pts) 2. By expressing the vectors T(a′ 1), T(a′ 2), T(a′ 3) in B′, write the matrix D = [T]B′. (2 pts) 3. Write the matrices [IdV ]B,B′ and [IdV ]B′,B. (2 pts) 1 4. What matrix would you obtain if you compute the product [IdV ]B,B′ · D · [IdV ]B′,B ? Justify your answer (do not evaluate the product array-wise in this question). (2 pts) 5. Check the result of the preceding question by computing explicitly the product of matrices [IdV ]B,B′ · D · [IdV ]B′,B. (2 pts) 6. Give a basis of Im T made of vectors expressed in B′. (2 pts) 7. Give a basis of Im T made of vectors expressed in B. (2 pts) Exercise 4 Let W be the vector subspace of R3 spanned by S = {b1, b2}, where b1 = (1, 0, 0) and b2 = (−2, 0, −1). 1. Check that S is a basis of W. (2 pts) 2. R3 is equipped with its standard Euclidian structure. Check that S is not an orthonormal basis of W. (2 pts) 3. Let L be the vector line spanned by b1, and pL : R3 −→ R3 the orthogonal projection on L. Find a formula giving pL(x) for any x ∈ R3. (3 pts) 4. Apply Gram-Schmidt process on S to construct an orthonormal basis {c1, c2} for the vector subspace W. (5 pts) Exercise 5 In the vector space P9(R) of polynomials whose degree is less or equal to 9, we consider the vector subspace W = P4(R). Find the rank and the trace of the linear operator TW : P9(R) −→ P9(R) which associates to every polynomial p defined by p(x) = a0 + a1x + ... + a9x9 the polynomial TW(p) defined by TW(p)(x) = a0 + a1x + a2x2 + a3x3 + a4x4. (3 pts) 2