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Unformatted text preview: Name: Signature: Student Number: Dr. J. Nikiel : Mathematics 218, Sections 1 and 2
Final Examination, June 7, 2003, 15:00—17:00 W Important Instructions:
1. This exam consists of two sets of problems. The ten “TRUE/ FALSE answer” questions are numbered A, B, . . . , J; each of them is worth 4 pts. The eight “workout” questions are
numbered 1, 2, . . . , 8; each of them is worth 8 pts. The maximum score is 104 points. 2. Do not separate these pages. 3. The pink booklets are merely a source of scrap paper. I will not read what is in them. W If you have any comments / requests write them here: M No calculators. Good Luck! Part I.
Write your answer TRUE or FALSE under each of the following problems / statements: W
Problem A. If det(A) = 2, then A is an orthogonal matrix. Your answer: m Problem B. If A and B are square matrices of the same size, then det(A + B) = det(A) + det(B). Your ans Wer: W Problem C. If A and B are symmetric matrices of the same size, then 0514 + £319 is a symmetric matrix for each choice of scalars a and Your answer: I l‘Jlia ' VS IVER L l u I: A R Y
m: usmm
Problem D. If 17 is an eigenvector of a square matrix A, then the same 17 is an eigenvector of A3. Your answer: Problem E. Let 271 = (1,1,1,1), 172 = (2,2,2,U), 173 = (3,3,0,0) and 174 = (4,0,0,0). Then S = {171,U2,173,274} is a basis for R4. Your answer: Problem F. If A is a matrix, then each vector 23' belonging to the null—space of A must be orthogonal to each vector 11'} belonging to the row space of A. Your answer: Problem G. If A = l is an eigenvalue of a symmetric matrix A, then A = 41 is an eigenvalue of A, too. Your answer: Problem H. If A and B are square matrices of the same size, then (AB)2 = A232. Your answer: Problem I. If a square matrix A has a row of 1’s or a column of 1’s, then det(A) 7E 0. Your answer: Problem 3. If {111,112, . . . ,uk} is an orthogonal set of vectors in an inner product space V such that dim(V) = n, then H S 16. Your answer: Part II. You must provide all essential details of your solution to each of the following problems
1, . . . ,8 on the page that contains the problem (continue on the reverse side of that page when needed). Problem 1.
1 2 0 0 2
1 0 3 2 1 _. w 1 ‘
Use Cramer s Rule to solve the system 2 l 0 2  a: = 0 Evaluate deter
1 1 0 1 1 minants in the denominators, but do n_ot evaluate determinants in the nominators. SHERMAN UNWER u‘ LIBRARY m IsliillU'r h Problem 2.
Find a matrix P that diagonalizes the following matrix A. Then ﬁnd P—lAP.
1 l 1
A = 0 2 2
0 0 3 Problem 3. Find the least squares solution of the linear system 1 1
2 2 §:'=
1 0 l
3 0 cu— mm r'\'1\"Eﬁ LHJi.;\RY OF HEIRUT Problem 4. Find an orthonormal basis for the row space of the following matrix ODJMH
Nb—‘P—‘H
Jackal—
HOOP4 EMERJFAN um " ‘ LIBRARY I
I. m' lHZlnu'; Let. A be a matrix of size 6 x 5. Suppose that the nullSpace of A is spanned by the vectors
171 = (1,1,1,0,0), 172 = (0,0,1,1,1) and '53 = (1,1,2, 1,1). Use this information to ﬁnd the
rank of A. Problem 5. Problem 6. Let A be a square matrix of size n X n. Write four conditions equivalent to “A is singular”. “H ' ‘ \‘vinsr’r L l l: u .\ R Y
0F BEIRUT Problem 7. Let V be an inner product space and W be a. subspace of V. Let T : V —> W denote the orthogonal projection of V onto W. Prove that ker(T) = Wt. .‘h “st.3 ; M
' ' "WEB
I L ‘ u i: a R Y
UP ﬁmRUT Let V1 and V2 be subspaces of R7. Let V = V1 ﬂ V2. Suppose that dim(V1) = 4 and
dim(V2) = 5. Prove that V must be inﬁnite. Problem 8. ...
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 Spring '07
 RanaNassif
 Linear Algebra, Algebra, Orthogonal matrix, Hilbert space, Det, Dr. J. Nikiel

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