MATH 218 Final, Fall 2007

American University of Beirut Fall 2007-2008 Elementary Linear Algebra (Math 218) Lecturer: Dr. Wassim Daher Section 3, MWF: 1:00 pm - 2:00 pm Final Exam, January 23 2008 Time: 2 hours The quality of your writing, the clarity and the precision of your reasoning, all together will be taken into account in the correction process. Open book and Calculators are not allowed for this Final Exercise 1.(10 Points) Let L : M22 → M22 be the linear operator defined by L(M) = M T Find the matrix for L with respect to the standard basis for M22 Exercise 2.(10 Points) Find (s, t) so    1 0 0 1 1 1    � s t � is close as possible to    1 1 0    Exercise 3.(10 Points) Find an orthogonal basis for the subspace w + 2x + 3y + 4z = 0 of R4. Exercise 4.(15 Points) Let A =    −5 3 4 −2 2 2 −6 3 5    a) Find the eigenvalues of A? Is A diagonalizable?. b) Find a basis to each eigenspace corresponding to the eigenvalues found in part a) and give its dimension. 1 c) calculate A5. Exercise 5.(15 Points) Consider the two subspaces of R3: E1 = {(x, y, z) ∈ R3, 3x − 2y + 3z = 0}, E2 = {(x, y, z) ∈ R3, x + 3y − z = 0 and 3x + 3y + z = 0} a) Find the bases B1 and B2 for E1 and E2 respectively. b) Prove that R3 = E1 ⊕ E2 i.e. 1) E1 ∩ E2 = {−→0 R3} and 2) R3 = E1 + E2 Exercise 6.(10 Points) Consider the bases for P1, B = {p1 = 6 + 3x, p2 = 10 + 2x} and B′ = {q1 = 2, q2 = 3 + 2x} a) Find the transition matrix from B′ to B b) Find the transition matrix from B to B′ c) Compute the coordinate vector [p]B, where p = −4+ x and then find [p]B′ Exercise 7.(20 Points) Let A =    2 −1 −1 −1 2 −1 −1 −1 2    a) Show that A is diagonalizable b) Find the matrix P that orthogonally diagonalizes A. c) Find an orthogonal basis for R3 Exercise 8.(10 Points) Let E be a finite dimensional vector space and let f be a linear operator from E to E. Show the following statement Ker(f) = Ker(f ◦ f) ⇒ Range(f) = Range(f ◦ f) Good Luck 2