MATH 218 Final, Spring 2014

May 12, 2014 MATHEMATICS 218 NAME ------------------- Time : 90 minutes Final Examination ID# ------------------ Spring 2013-14 Circle your section number : Sabine El Khoury Michella Bou Eid Monique Azar Hazar Abu-Khuzam 1 2 3 4 5 6 7 8 9 10 11 12 9 M 2 F 8 M 1 W 2 F 1 M 3:30 T 5 T 12:30 T 1 F 11 M 11 F PROBLEM GRADE PART I 1 ------- /16 2 ------- / 18 3 ------- / 10 4. ------- / 12 5. ------- / 7 PART II 6 7 8 9 10 11 12 a a a a a a a b b b b b b b c c c c c c c d d d d d d d 6-12 ------- / 21 PART III a b c d e f g h 13 ------- / 16 TOTAL ------- /100 1 PART I. Answer each of the following problems in the space provided for each problem ( Problem 1 to Problem 5). 1. Let A=           6 0 0 0 5 1 0 1 5 (a) Find the eigenvalues and a basis for each eigenspace of A. [ 10 points] 2 1(b) Show that A is diagonalizable. Find an invertible matrix P and a diagonal matrix D such that P-1AP=D. ( Do not verify) [ 6 points] 3 2. Let T : P2 → P2 be the linear transformation defined by ( ) ( ( )) xp x T p x ′ = . (a) Find the matrix A= of T relative to the standard ordered basis β ={1, x, x2} of P2 . [6 points] (b) Find the matrix B= of T relative to the ordered basis β’={1+x2, 2x, 1} of P2. [6 points] 4 (c) Find the transition matrix P=[ ] β I β ' from β’ to β such that B=P-1AP ( Do not verify). [6 points] 3. Prove that an orthogonal set of nonzero vectors in an inner product space V is linearly independent. [ 10 points] 5 4. Let T: R3 → R3 be the linear transformation defined by T           c b a =           + + 0 0 c b a . Find a basis for the null space N(T) and use the Gram-Schmidt process to construct an orthonormal basis for N(T), with the usual dot product. [12 points] 6 5. Let T: R3 → R4 and S: R4 → R2 be two linear transformations such that the composition SοT =0. Show that if S is onto then T cannot be one-to-one. [7 points] 7 PART II. Circle the correct answer for each of the following problems ( Problem 6 to Problem 12). IN THE TABLE IN THE FRONT PAGE [3 points for each correct answer] 6. Let T : R3 → R3 be the linear map defined by T           z y x =           − + 0 2 3 x y x . Then dim ( Range T) is : a. 3 b. 2 c. 1 d. none of the above. [ 3 points] 7. Let S be the subspace defined by S={M is a 3x3 skew-symmetric matrix with the sum of the entries of each row is zero}. Then dim S= a. 1 b. 2 c. 3 d. none of the above. [ 3 points] 8. If T: R2→ R is a linear transformation such that T(v)≠0 for some v in R2, then a. T is one-to-one. b. T is onto. c. Dim (Nullspace T)=0. d. none of the above. [ 3 points] 8 9. Let A=           1 2 2 1 1 2 and b=          − 1 0 2 . Then the least squares solution to Ax = b is: a. xˆ =       /5 6 /5 3 b. xˆ =      − /5 3 /5 6 c. xˆ =      − /5 6 /5 3 d. None of the above [ 3 points] 10. Let U be the subspace of R3 defined by 0 . 3 2 3 |       = + − ∈       = z y x R z y x U Then dimU ⊥ = a. 3 b. 1 c. 2 d. none of the above [ 3 points] 11. Let A=           3 1 0 8 1 0 2 1 1 . Then: a. A is diagonalizable. b. A is not invertible. c. The eigenspace corresponding to the eigenvalue λ=1 has dimension 2. d. None of the above. [ 3 points] 12. Let T: V→V be a linear transformation with dim V= n such that T is onto. Which one of the following statements is FALSE: a. T is an isomorphism b. If { v1, v2,…, vn} is linearly independent in V, then {T(v1), T(v2), …, T(vn)} is linearly independent in V. c. dim( RangeT)=n d. none of the above. [ 3 points] 9 PART III. Answer TRUE or FALSE only IN THE TABLE IN THE FRONT PAGE ( 2 points for each correct answer) a. ----- If A is a 3×3 matrix such that A2=0, then rank A ≠3. b. ----- Let V be a finite dimensional vector space and let W be a subspace of V. If dim W= dim V, then W=V. c. ----- If A is a 2×5 matrix, then dim (Column space of A) ≤ 2 d. ------ Let V be a finite dimensional inner product space and let W be a subspace of V. Then the orthogonal complement of W ⊥ is equal to W. e. ------- Let V be a finite-dimensional vector space and let T: V→V be a linear transformation. If T is one-to-one, then T is onto. f. ----- The matrices A=       − 4 3 2 1 and B=      − 3 4 1 2 are similar. g. ---- Let T: R2→ R3 be a linear transformation such that T       0 1 =           1 1 1 , and T       1 0 =           − 3 2 1 . Then T       −1 5 =           − 2 7 4 . h. ----- The set of all 2×2 noninvertible matrices is a subspace of M2×2. [16 points]. . 10 11