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LIBRARY OF "E'RUT
American University of Beirut MATH 218
Elementary Linear Algebra
Spring 2004 Final Exam Date: Wednesday, June 9, 2004  11:30 am to 1:30 pm
Instructor: Dr. Mohamed Kobeissi This is NOT an open‘book exam. Your exam should have 1 1 pages including this one, and there
are 6 questions totaling 100 points. You can continue each exercise on the reverse side of the paper if needed. Question Grade
1 . Good luck Exercise 1 Let A = a. ( 7 points) For what value of k the matrix A is invertibie? b. (3 points) For k = 2, solve the system AX I Exercise 2 Lets: = [1,—131,1),y : (—1,—1,1,—1):z : (0,1,110),and1et1/V :span{x,y=z}
be a subspace of R4. N a. (5 points) Show that g is an orthogonal set of vectors relative to the Euclidean inner product. b. (3 points) What is the dimension of 14/"? Justify. c. (8 points} Find a basis for W”: the orthogonal complement of WC d. (4 points) Find a vectorf 6 1R“ such that the {an y, z, 1:} form a basis of R4. Justify. Exercise 3 (10 points) Let W be the subspace of R3 spanned by {(11 1, 2). (1, ——1, 1)}, and let
I b = 1
0 Find projwb, the orthogonal projection of b on W. GOOD 1 2
Exercise 4 Consider the matrix A : 0 0
2 1 a. (4 points) Find the characteristic equation of A. b. (8 points) Find the eigenvalues of A, and a basis for each eigenspace of A. c. (10 points) Find an orthogonal matrix P that diagonalize A, and Show the relation between
A, P. P”1 and D. d. (3 poinrs) What is the rank of A? Justify. Exercise 5 In an inner product space V, prove the following: a. (5 poinrs) < 145,?) >= 41”” + 'UHZ — iHU _ ”H2 13. (5120mm) Hu+vll s Hull+ llvH (him: you may use the CauchySchwarz inequality, < 11.32," >  S Hu‘LHLH) Exercise 6 (251)0inrs, 5 points each) Prove (concisely) or disprove (by a counter example), the
following statements. a. Let A be a n >< n matrix. i) If A is invertible, then the orthogonal complement of the row space of A is reduced to
0. ii) If 0 is an eigenvalue of A4, then A is invertible. iii) If A is an eigenvalue of A with corresponding eigenvector 1', then A3 is an eigenvalue of .43 with the same corresponding eigenvector. b. The Nullity of a 3 X 4 matrix A can be equal to zero. (3. If a set of 3 vectors in R3 have the property that each 2 of them are linearly independent,
then this set is linearly independent. ...
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 Spring '07
 RanaNassif
 Linear Algebra, Algebra

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