NO. 2 week5-week7 Student Full Name:___________________________________ . Student ID:__________________________________________ . CRN No:____________________________________________ . Branch: _____________________________________________. Linear Algebra (Math-251) Total Points True/False ____/6 MCQ ____/6 Short Answer ____/18 Total ____/30 Due Date Date: 24-03-2017 Max Marks: 30 Section-I State whether the following statements are true or false: [6X1=6] 1. Vectors (2,1,3), (2, 1, 3) are orthogonal to each other. (1)........ 2. If A is matrix then row space of A and column space of A have same dimension. (2)........ 3. T: 2 , (, ) = 6 7 is a linear transformation. (3)........ 4. If is a 3 3 invertible matrix then the columns of form a basis for 3 . (4)....... 5. The set = {(1,2), (6,3)} is linearly independent. (5)....... 6. If is a finite set of vectors in a vector space and a basis for then it must be linearly dependent and spans . (6)....... Section-II For Each Question, Choose the Correct Answer from the Multiple-Choice: [6X1=6] 1. If u = (3, 2,1) and v = (4, 1,5) are two vectors in R3 . Then the cross product u v is a. (12,2,5) b. (7, 3,6) c. (0,0,0) d. (9, 11,5) 2. If = (4,2, 4) and = 18 . Then the value of is: a. 1 b. 3 c. 6 d. 2 3. Let S = {v1 , v2 , v3 } is a basis of V and v= 2v1 + 4v2 v3 . Then the coordinate vector of v relative to S; ( (v)s ) is : a. (1, 2,4) b. (2,4, 1) c. (2,4) d. (2, 4,1 4. A linear combination formed by the vectors w = (7,2,6) , v = (3,1,4) and u = (1,0,2) is: a. = 3 3 b. = + 2 c. = 2 d. + + = 0 4. If = (1,4,4) and = (3,2,6). Then the Euclidean distance (, ) = a. 35 b. 12 c. 12 d. 35 6. If A is matrix, then: a. rank() = b. rank() = + c. rank() = d. rank() min(, ) Section-II (Multiple Choice Questions) MCQ 1 Answers 2 3 4 5 6 Section-III Attempt all the questions: [6X3=18] 1. If = (4,2, 4) and = (6, 12, 4). Find , where is the angle between and . + 3 2. Let W = {[ 5 ] , | , } be a subset of 3 . Find a 4 2 basis for W. 3. Let be the set of all vectors of the form (, 1,1). Check if W is a subspace of 3 or not. 4. Let (x, y) = (x, 2y, 3x y), find the domain, codomain and the image of (4, 1). 5. Let 1 = (3,0, 6), 2 = (4,1,7) and 3 = (2,1,5). Determine if {1 , 2 , 3 } is a basis for 3 . 6. Given the 3 4 matrix. Find the basis for Row space of A and its dimension where 1 A=[ 3 2 4 7 0 2 3 6 1] 1 20