Vector Calculus and Linear Algebra — Winter 2022

Objective Questions (MCQ) 1. If Ais invertible n× n matrix, then for each n×1 matrix b, the system of equations Ax = b has _____solution.

Infinitely many (b) no solution (c) exactly one (d) none of these 2. If u and v are vectors in Rn then the Cauchy-Schwarz inequality is__

|u.v| ≤ ‖u‖‖v‖ (b) |u.v| ≥ ‖u‖‖v‖

|u.v| ≠ ‖u‖‖v‖ (d) none of these 3. 1 0 The eigen values of the matrix [ ] are 2 −1

2,−1 (b) 0,2 (c) 1,0 (d) 1,−1 4. The set s = {(0,1, 0),(1,0,1),(1,0,−1)} is an orthogonal set of vectors in an inner product space then the set s is ____

linearly independent (b) linearly dependent

both (d) none of these 5. If the matrix Aof order n is orthogonally diagonalizable then Ais

symmetric (b) skew symmetric (c) unitary (d) Hermitian 6. The vector V = (x+3y)i+(y−2z)j+(x+𝜆z)k is solenoidal for the value of 𝜆 = ____

2 (b) 0 (c) −2 (d) −1 7. If 𝜙 = x +y+z then the value of |grad 𝜙| is

√3 (b) 0 (c)3 (d)

Objective Questions (MCQ) 1. The rank of the matrix whose every element is unity, is __________

Zero (b) Greater than one (c) 2 (d) Equals to one 2. The set {(1,0),(1,1)} is____

linearly dependent (b) linearly independent

Basis of R2 (d) none of these 3. If 2,3,4 are eigen values of the matrix A, then the eigen values of AT are________

1/2,1/3,1/4 (b) 2,3,4 (c) 4,9,16 (d) Zero 4. T:V → Vis a linear operator and Vis a finite dimensional vector space, then Tis one-to-one if

R(T) = V (b) nullity(T) ≠ 0 (c) ker(T) ) ≠ 0 (d) none of these 5. The vectors u = (2,k,6) and v = (1,2,3) are orthogonal with respect to Euclidean inner product, then the value of k = _____

10 (b) −5 (c) zero (d) −10 6. If r = xi+yj+zk the ∇.r = ____

0 (b) 3 (c) 1 (d) −3 7. The field Fis conservative on Dthen the value of ∫F ⋅dr around every closed loop in Dis____

zero (b) −1 (c) none of these (d)

Find the coordinate vector of (2,−1,3) relative to the basis s = {v ,v ,v }, where v = (1,0,0 ), v = (2,2,0 ) and v = (3,3,3 )

1 0 1 Find the inverse of the matrix A = [0 1 1] by using Gauss Jordan110 elimination method. Investigate for which values of a, the system of equations

07 x +x +x = 1,x +2x +4x = a,x +4x +10x = a2 has no solution, unique solution or infinitely many solutions.

Find the kernel of the linear operator T:R2 → R2 defined by T(x,y) = (x−y, x −y) and determine whether the linear transformation Tis one-to- one.

Consider the bases S = {v ,v } for R2, where v = (−2,1) and v = (1,3) and let T:R2 → R3 be the linear transformation such that T(v ) =1 (−1,2,0) and T(v ) = (0,−3,5). Find a formula for T(x ,x ) and use that212 formula to find T(2,−3).

Determine whether the set of all pairs of real numbers (x,y) with the operations (x,y)+(x′,y′) = (x+x′,y+y′) and k(x,y) = (2kx,2ky) is a vector space.

Show that the set of functions {1,ex,e2x} form a linearly independent set of vectors in C2(−∞,∞).

Determine whether the vector (7,8,9) is a linear combination of vectors u = (2,1,4),v = (1,−1,3) and w = (3,2,5). 0 0 −2

Find a matrix Pthat diagonalizes A = [1 2 1 ] and determine P−1AP.103

Determine whether the transformation T:R2 → R2defined as T(x,y) = (x+3y,3x −y) is linear or not?

Find the least squares solution of the linear system Ax = b given by x −1 x = 4,3x +2x = 1,−2x +4x = 3 and find the orthogonal projection of b on the column space of A.

Let R3 have the Euclidean inner product. Use Gram-Schmidt process to transform the basis {(1,1,1),(−1,1,0),(1,2,1)} into an orthonormal basis.

x3 x2 Find the line integral of f(x,y) = over the curve C: y = , 0 ≤ x ≤ 2. y

How much work is required to move an object in vector force field F = 〈yz,xy,xz〉 along path r(t) = 〈t2,t,t4〉,0 ≤ t ≤ 1 ?

Define curl and divergence of a vector field. The vector field is given by A = (x2 +xy2)i+(y2+x2y)j. Show that the field Ais irrotational and also find the scalar potential.

Use the inner product 〈f,g〉 = ∫ 1 f(x)g(x)dx; to compute d(f,g) and ‖f‖, where f(x) = x, g(x) = ex in C[0,1].

Find the directional derivative of the function 𝜙 = xy2 +yz2 at point P(2,−1,1) along the tangent to the curve x = atsint, y = atcost, z = at at t = 𝜋 ⁄ .4

Verify Green’s theorem for ∮[(xy+y2)dx+x2dy] where Cis bounded by c y = x and y = x2