Vector Calculus and Linear Algebra — Summer 2019

Objective Question (MCQ)

07 1. The matrix is in the form

Row (b) Reduced row (c) Both (a) and (b). (d) None. echelon. echelon. 2 . For the | A k | =_____

1 (b) 2 (c) 2k (d) 2k-1 3. If u and v are vectors in a real inner product space, and ||u||=2, ||v||=3, then |<u,v>| ≤ ______

6 (b) 3 (c) 2 (d) 1.5 4. Which of the following doesn’t lie in the space spanned by cos2x and sin2x ?

1 (b) 0 (c) Sin x (d) Cos 2x 5. Dimension of the subspace { p(x) P : p(0) = 0 } of P ={a+bx+cx2 : a, b, c R} is22

3 (b) 2 (c) 1 (d) 6. Which of the following subsets of R2 is linearly dependent?

{(1,2), (2,1)} (b) {(1,2), (2,1), (1,1)} (c) {(1,2)} (d) None 7. Let T: R2 R2 defined by T(x,y) = (x,0) then Ker (T) =____

Y-axis (b) X-axis (c) Origin (d) None

07 1. Which of the following is not an elementary matrix?

(b) (c) (d) 2. For = (1, −1, 2) , = (1, 3, 1) are vectors of R3 with Euclidean inner product then cos =______, where is the angle between the two vectors.

1 (b) 0 (c) − 3 (d) 3. Which of the following is not true?

(AB)T = BTAT (b) (AB)−1 = B−1A−1 (c) (AT)T = A (d) AT= −A 4. If Ais n×n matrix having rank n−1 then A, A2,A3, ……….., Ak,…. have common eigenvalue _____

1 (b) −1 (c) 0 (d) 5. If Ais unitary matrix then A−1 =____

A (b) A2 (c) AT (d) I 6. The dimension of the solution space of x − y = 0 is ____

0 (b) 1 (c) 2 (d) 7. If f(x,y,z) = xyz then Curl ( grad f ) = ___

0 (b) x (c) xi+yj+zk (d) xyz

Which of the following are linear combination of u = (0, −2, 2) and v = (1, 3, −1)? Justify! (i) (2,2,2), (ii) (0, 4, 5)

Using Gram-Schmidt orthogonalization process find the corresponding orthonormal set to { (1, 1, 1), (0, 1, 1), (0, 0, 1)}.

Using Gauss- Jordan elimination find the inverse of .

Find the rank of the matrix and basis of the null space of .

Solve the system of linear equations using Gauss elimination method: x + y + 2z = 8, − x – 2y + 3z = 1, 3x – 7y +4z = 10.

Show that the set of all real numbers of the form (x, 1) with operations (x, 1) + (x’, 1) = (x + x’, 1) and k(x, 1) = (kx, 1) forms a vector space.

Determine whether the following are linear transformation or not?

T: P P , T(p(x)) = p(x + 1),22 (ii) T: P P , T(a + bx + cx2) = (a + 1) + (b + 1)x + (c + 1) x2.22

Which of the following sets of vectors of R3 are linearly independent? Justify.

{ (4, −1, 2), (−4, 10, 2)} (ii) {(−3, 0, 4), (5, −1, 2), (1, 1, 3)}

Find the eigenvalues and bases for the eigenspaces for A11 ,

Find basis of kernel and range of T: R2 R2 , defined by T( x, y) = ( 2x − y, −8x + 4y)

Which of the following are basis of R3? Justify!

{ (1, 0, 0), (2, 2, 0), (3, 3, 3) } , (ii) { (3, 1, −4), (2, 5, 6), (1, 4, 8)}

Let T: P P , defined by T(p(x)) = p(3x − 5)22

Find the matrix of Twith respect to the basis {1, x, x2}. (ii) Use the indirect procedure using matrix to compute T(1 + 2x + 3x2). (iii) Check the result in (b) by computing T(1 + 2x + 3x2) directly.

Show that is irrotational.

Find the directional derivative of f(x, y, z) = x2z + y3z2 –xyz at (1,1,1) in the direction of the vector (−1,0,3).

07 Using Green’s theorem evaluate (3x2 − 8y2) dx + (4y – 6xy) dy, where Cis the boundary of the region bounded by y2 = x and y = x2.

Find the work done by = (y – x2) i + (z – y2) j + (x – z2) k over the curve r(t) = t i + t2 j + t3 k; , from (0,0,0) to (1,1,1).

Use Cramer’s rule to solve: x + 2z = 6, – x + 4y + 6z = 30, – x – 2y + 3z = 8.

Verify divergence theorem for = x i + yj + zk over the sphere x2 + y2 + z2 = a2.