MATH 218 Final, Fall 2011

MATHEMATICS 218 FALL SEMESTER 2010-2011 FINAL EXAMINATION Time: 120 minutes Date: January 31, 2011 Name:—————————————————————————- ID Number:———————————————————————– Section:————————————————————————– QUESTION GRADE 1 /25 2 /20 3 /25 4 /30 5 /30 6 /40 7 /30 TOTAL GRADE /200 Answer the following set of questions on the allocated pages; the back of pages may be used if needed 1 2 1. Let A =    0 0 −2 1 2 1 1 0 3    (a) Find the eigenvalues of A. (9 points) (b) Find a basis for each of the eigenspaces. (10 points) (c) Deduce that A is diagonalizable and write a formula calculating A5. (6 points) 3 4 2. Consider the following system:      x + 2z = 2 2x + 4z = 6 y + 2z = 4 (a) Find the least squares solution of the above system Ax = b (12 points) (b) Find the projection of b onto the Column Space of (A). (8 points) 5 6 3. Let W =            x y z w   ∈ R4 | w + 2x + 2y + 4z = 0          (a) Find a basis for W. (7 points) (b) Find a basis for W ⊥. (8 points) (c) Use parts (a) and (b) to find an orthogonal basis for R4 with respect to the Euclidean inner product. (10 points) 7 8 4. Let T : P3 → P3 be the linear transformation defined by: T(p(x)) = p′(x) − p”(x) + p(0) (a) Let B = {1, x, x2, x3}, find [T]B (6 points) (b) Find a basis for the nullspace of T. (8 points) (c) Find a basis for the range of T. (6 points) (d) Let B′ = {1, x − 1, x2 − 2x, x3}, use (a) to find [T]B′. (10 points) 9 10 5. Answer the following INDEPENDENT questions. (10 points each) (a) • For which values of t is < p, q >= � 2 t p(x)q(x)dx an inner prod- uct on V = P2? • For which of the values of t obtained above would the two poly- nomials p(x) = x + 1 and q(x) = 2 be orthogonal? 11 (b) Let W be a subspace of V = M2×2, with basis B = {I, A}, where I = � 1 0 0 1 � and A = � 8 4 2 1 � . Let T : W → W be given by T(M) = AM. • Find [T]B • Use [T]B to compute T(3I − A). 12 (c) For which value(s) of k would the vector v = � 1 −1 � be an eigenvector of the matrix A = � k 0 1 −1 � ? 13 6. Prove the following: (10 points each) (a) Let W = span {w} be a subspace of an inner product space V . Let b ∈ V and not in W. Use the Cauchy-Schwartz inequality to prove that: ||projW(b)|| ≤ ||b|| 14 (b) Let V = P2 and p, q ∈ P2. Show that: < p, q >= p(0)q(0) + p(1 2)q(1 2) + p(1)q(1) is an inner product in P2. 15 (c) Let A be an n × n matrix. • Show that if A2 = 0 then Column space of (A) ⊂ Nullspace (A) • Deduce that if A2 = 0 then rankA ≤ n 2 16 (d) Let T : V → V be a linear transformation such that T ◦ T = T. Show that Range(T) ∩ Nullspace(T) = 0 17 7. Indicate whether each of the following statements is TRUE (T) or FALSE (F) without justifying your answer. (3 points each) —-(a) The set of all 3 × 3 matrices with det(A) = 0 is a subspace of M3×3. —-(b) Let T : P2 → R defined by T(ax2 + bx + c) = |a| is a linear transformation. —-(c) Let T : V → W be a linear transformation. If dimV < dimW then T cannot be onto. —-(d) Let A, B and C be n × n matrices, if AB = AC then B = C. —-(e) The dimension of the vector space of the n×n upper triangular matrices is n2−n 2 . —-(f) T : V → W be a linear transformation with {v1 . . . vn} linearly independent in V , then {T(v1) . . . T(vn)} are linearly independent. —-(g) If λ is an eigenvalue of an idempotent matrix A ( i.e: A2 = A) then λ = 0 or λ = 1. —-(h) In R2 with the standard inner product, the orthogonal com- plement of y = 2x is y = 1 2x. —-(i) If T : Rn → Rm is a linear transformation and m = n then T is an isomorphism. —-(j) If λ is an eigenvalue for A then λn is an eigenvalue for An for all n ∈ Z.