Vector Calculus and Linear Algebra — Summer 2018

Objective Question (MCQ) Mark

Choose the appropriate answer for the following questions. 1. Asquare matrix whose determinant is non zero is called (A) Singular (B) non-singular (C) invertible (D) both Band C 2. If Aand Bare non singular matrices then ( A B )  1  _ _ _ _ (A) A  1 B  1 (B) A B (C) B  1 A  1 (D) none of these 3. If A     1     then Ais in (A) Row echelon form (B) Reduced Row echelon form (C) both A and B (D) none of these 4. For what values of k does the system x  y  2 , 3 x  3 y  k has infinitely many solutions (A) K=5 (B) k=4 (C) k=6 (D) k=1 5. If in a set of vectors atleast one member can be expressed as a linear combination of the remaining vectors then the set is (A) Linearly independent (B) Linearly dependent (C) basis (D) none of these 6. If Vis any vector space and Sbe a subset of Vthen Sis called basis for Vif (A) Sis Linearly independent (B) Sspans V (C) both Aand B (D) Sis Linearly dependent 7. For what value of k the vectors u and v are orthogonal where u=(2,1,3) , v=(1,7, k) (A) K=-3 (B) k=1 (C) k=5 (D) k=2

Choose the appropriate answer for the following questions. 1. 1 0 The eigen values of a matrix A  are  24   (A) 1,4 (B) -1,-4 (C) 1,3 (D) -1,3 2. If Ais a nxn size invertible matrix then rank of Ais (A) n-1 (B) n (C) 2n (D) n+1 3. If Fis solenoidal then (A)  F  0 (B)   F  0 (C)   F  0 (D) none of these 4. The mapping T : R 3  R 3 d e f i n e d b y T ( x , y , z )  ( x , y ,  z ) is called as (A) Contraction (B) Projection (C) Reflection (D) Rotation 5. The linear transformation T : V  Wis one to one if and only if the nullspace of Tconsists of only (A) Identity vector (B) zero vector (C) any non zero vector (D) none of these 6. If A   1   then the rank of the matrix Ais (A) 1 (B) 2 (C) 0 (D) 7. Let Abe a skew-symmetric matrix then (A) a ij  a ji (B) a ij   a ji (C) a ii  0 (D) both Band C

Find the unit vector normal to the surface x y 3 z 2  4 a t (  1 ,  1 , 2 )

Express the matrix A     3 7 6    04 as the sum of a symmetric and skey-symmetric matrix.

Investigate for what values of a n d   the equations 2 x 3 y 5 z 9 , 7 x 3 y 2 z 8 , 2 x 3 y z           07 have (1) No solution (2) a unique solution (3) infinite number of solutions

Find the rank of the matrix    1    03

04 2 3 4   Find the inverse of the matrix 4 3 1 by Gauss Jordan Method   1 2 4  

For the basis S   v1 , v2 , v3  o f R 3 where v  (1,1,1),v  (1,1,0),v  (1,0,0) Let123 T : R 3  R 207 be the linear transformation such that T (v )  (1,0),T (v )  (2,1),T (v )  (4,3) find a formula123 for T (x , x , x ) and then use the formula to find T (4,3,2)123

Determine whether the vector v  (  5 , 1 1 ,  7 ) is a linear combination of the vectors v1  ( 1 ,  2 , 2 ) , v2  ( 0 , 5 , 5 ) , v3  ( 2 , 0 , 8 )03

Solve the linear system x  y  z  4 ,  x  y  z   2 , 2 x  y  2 z  204 by gauss elimination method.

Let R 3 have the Euclidean inner product. Use the gram schmidt process to transform the basis ( u1 , u2 , u3 ) in to Orthonormal basis where u1  ( 1 , 0 , 0 ) , u2  ( 3 , 7 ,  2 ) , u3  ( 0 , 4 , 1 )07

Find the eigen values and corresponding eigen vectors of 2 12 A    1 5  

Let A      1  3     a n d b      1    04 then find the least squares solutions to AX=b

Let T : R 3  R 3 be a linear operator and B  ( v1 , v2 , v3 ) a basis for R 307 . Suppose that T (v )  (1,1,0),T (v )  (1,0,1),T (v )  (2,1,1) then123 (1) Is (1,2,1) in R(T) ? (2) Find a basis for R(T).

Find the work done by the force F  (3x 2  3x)i  3zj  k along the straight line t i  t j  t k , 0  t  1 .03 .

Check whether the vectors (2,-3,1), (4,1,1),(0,-7,1) is a basis for R

Verify Green’s Theorem for F  ( x  y ) i  x j a n d Ci s x2  y2  107

Find the directional derivative of 4 x z 2  x 2 y z a t ( 1 ,  2 ,  1 ) 03 in the direction of 2i  j  2k

Show that F  ( e x c o s y  y z ) i  ( x z  e x s i n y ) j  ( x y  z ) k is conservative and find the potential function.

Let V  (a,b) / a,b  Rand let v  (v ,v ),w  (w w ) 1 2 1, then define ( v1 , v2 )  ( w1 , w2 )  ( v1  w1  1 , v2  w2  1 ) and c ( v1 , v2 )  ( c v1  c  1 , c v2  c  1 )07 then verify that Vis a vector space.