NO. 3 week9-week11 Student Full Name:___________________________________ . Student ID:__________________________________________ . CRN No:____________________________________________ . Branch: _____________________________________________. Linear Algebra (Math-251) Total Points True/False ____/6 MCQ ____/6 Short Answer ____/18 Total ____/30 1|7Page Due date: 05 May 2017 Max Marks: 30 Section-I State whether the following statements are true or false: [6X1=6] 1. Matrix is diagonalizable if it is similar to a diagonal matrix B; that is, there exists an invertible matrix and = 1 . (1).............. 1 0 2. The eigenvalues of the matrix A =[2 3 0 4 7. 0 0 ], are 1, 3, and 7 (2).............. 3. A square complex matrix is called Unitary if its conjugate transpose equal to the matrix . (3).............. 4. The inner product of a nonzero vector with itself ( u, u ) is always a positive real number. (4).............. 5. If = (1,2, 3,4) and = (2,1,4,3) then , = 0 , where , denotes the Euclidean inner product. 4 6. The matrix A = [2 2 (5).............. 2 2 3 3 6 ] is Hermitian. 6 3 2 (6).............. 2|7Page Section-II For Each Question, Choose the Correct Answer from the MultipleChoice: [6X1=6] 1. Which of the following sets of vectors are orthogonal with respect to the inner product, defined by u, v = 21 1 + 52 2 on R2: a. (2,5), (8,-3) b. (3,4), (2,6) c. (5,3), (3,-2) d. (1,5), (3,-2) 2. The values of for which = (, 7, 2), and = (, , 5) are orthogonal in R3 with respect to Inner Product u, v = 1 1 + 2 2 + 3 3 , are a. 2, -5 b. -2, 5 c. -2,- 5 d. 2, 5 3. If 0 is an eigenvalue of a square matrix A then A is: a. an Identity matrix. b. invertible. c. not invertible. d. None 4 + 2i 1 + 8 4. If V = [ 2 2i 1 3i] , a complex matrix, then is 8 6 7 5 4 2i 1 8 a. [ 2 + 2i 1 + 3i] 8 6 7 5 4 2i 2 + 8 b. [ 1 2i 6i 7] 8 1 + 3 5 4 2i 2 8 c. [ 1 + 2i 7 6i] 8 1 + 3 5 4 + 2i 1 + 8 d. [ 2 2i 1 3i]. 8 6 7 5 3|7Page 5 5. The characteristic equation of the matrix A = [ 4 a. 2 2 19 = 0 b. 2 + 2 19 = 0 c. 2 2 + 19 = 0 d. 2 + 2 + 19 = 0 1 is 3 6. Cosine value of the angle between vectors u and v is 1 and |||| = 3, |||| = 5, then u, v = a. 10 b. 15 c. 15 d. 10 Answer: 1 2 3 4 5 6 Section-III Attempt all the questions: [6X3=18] 1. Find the complex dot product , such that = (5, 2, 2 + , 4), = (3, 1 + 2, 1,3). Solution: 4|7Page 2. Find all least square solutions and the error vector of the linear system: 1 2 = 2 1 + 2 = 4 21 + 2 = 8 Solution: 3. Show that matrix A = 1 2 2 3 2 3 1 3 2 3 2 3 2 3 1 3 3 [3 Solution: 5|7Page is orthogonal and find A-1 4. Find the eigenvalues and the eigenvectors of the matrix 7 4 =[ 5 2 Solution: 5. Show that matrix + is Hermitian , where = 0 0 3 1 2 [ 2 0 ] , = [ 0 1 1] . 2 0 1 2 1 1 Solution: 6|7Page 6. Verify the property u + v, w = u, w + v, w , If the inner product on 2 , defined by u, v = 21 1 + 32 2 . Solution: 7|7Page