MATH 219 Final, Fall 2001

LIBRARY AMERICAN UNIVERSITY OF KEIRU? January 31, 2002 MATHEMATICS 219 Time : 2 Hours. FINAL EXAMINATION Prof. H. Abu-Khuzam FALL 2001-2002 PART IJ. Answer each of the following problems in the space provided for each problem( Problem { to Problem 6). . Let T:P,(R) > R? be the function defined by T(atbx+cx”)=(a, atb, 2a-b) (a) Show that T is a linear transformation. MERICAN CNIVERSIT LIBRARY OF BREA (b) Find Ker T. Is T an isomorphism? AMERICAN UNTVERS! LIBRARY OF RYERE? 01 1 2. LetA=}1 0 -1]. 00 1 (a) Find the eigenvalues of A [ 3 points] (b) Find bases for the corresponding eigenspaces of A. [ 6 points] 2 2. (c) Deduce that there is a basis of R® consisting of eigenvectors of A and find an invertible matrix P and a diagonal matrix D such that P'AP=D. ( Do not verify) [ 3 points] 3. Let T: R' > R°* be the linear transformation defined by T(x, y, A= (2y+z. x-4y, 3x) (a) Find the matrix A ol T relative to the standard ordered basis 8 = { (1,0,0), (0,1,0), (0.0. }or R’. [ 3 points] ~ 3 3. (b) Find the matrix B of T relative to the ordered basis 8’={ (0.0.1), (1,1,0), (1,0,0) of R. { 4 points} 3. (c) Find the change of basis matrix Q from 8’ to B such that B=Q'' AQ. (Do not verify) [ 3 points] 4. (a) Show that an orthogonal set of sonzero vectors in an inner product space V is linearly independent. [6 points] (b) Let V be the subspace 0! i! with basis { (1,2,0), (0,1,1)}. Use the Gram-Scmidt process to find an orthonormal basis of V. [ 4 points] 5. Let U={ p(x) € Pal RY} p'(0) = 0, and p’(x) =0 }. (a) Find a basis for the subspace 0 of PCR). [ 4 points] (b) Use linear extension to find a linear transformation T :P.(R) 3 P4(R) such that T(p(x)) = p(x) for all pox) € U .and T(x+1)= 0. { 6 points] 6. (a) Let A and B be simifar nn matrices such that A is invertible, Prove that B must be invertible. (4 points] (b) Suppose that 4 is an eigenvalue of an nxn matrix A. Show that 2 is an eigenvalue of the matrix A> {6 points] PART II. Circle the correct answer for each of the following problems ( Problem 7 to Problem 16). {4 points for each correct answer, 0 for no answer. and —! tor each wrong answer]. 7. Consider tie matrix 6) QO 0 A=|~5 3 Of. then 3. -4 1 Rank A =3, A is invertible. A is diagonalizable. —3 is an cigenvalue of A. [ 4 points] 8. If A is a 3x3 matrix such that |A|=4, Then | A (adj A)] is equal to: aot b. 16 c 64 da. none of the above [ 4 points] 9. The value of the number ¢ for which the matrix Q 0 2 is nob invertible is: none of the above [ 4 points] 10. Let T:R*—> R® be the linear transformation defined by TX. 2)=(K42y. \. 8. X-y) Then dim (Im T) is: aw 3 b. 2 a 4 a. none of the above. [ 4 points] Hl, Let S be the space of all skew-symmetric 4x4 matrices ( A's - A). Then dim S is equal to: a 10 b. 12 c 6 d. none of the above. (4 points] 12. WT: Vo Wis a linear transtormation such that the image of every linearly independent subset of V is linearly independent in W. then which one of the following statements is (ruc? a. Tis onto, b. Tis one -to -pne. ce. TéAd=0 for all vectors Ain VY. do none of the above. [ 4 points] 13. The subspace of R* spanned by 1.-2.3.5). (0.3.1-5). 14.0), and (0.0.0.1) has dimension: a 2 b 3 c 4 d. none of the above. L 4 points} 9 4. The value(s) of k for which the system Xt y+ 7=k X+2y+ 2753 y+ kes has no solution are: a. b. Let V and W be finite dimensional k=l k=0 k at hone of the above. { 4 points] which one of the following skuements is true? T is one-to-one = Tis onte dim (Ker T) + dim (lm T: = dim W {if dim V= dim W. then T is an isomorphism of V onto W. none of the above. ( 4 points] 16. Let A be an invertible nxn mairix over R. Which one of the following statements is FALSE? a. bh. c. d. Rank A= n. A! is invertible. The number of nonzero rows ia te row echelon form of Ais n The rows of A are lincarly dependent in R’, [ 4 points] tor spaces. and T: V->W is a linear transformation, Then,