Vector Calculus and Linear Algebra — Winter 2018

Objective Question (MCQ) Marks

07 1. For a square matrix , if , then is

Symmetric (b) Skew-symmetric (c) Orthogonal (d) Singular 2. For inner product defined by , value of is

(b) (c) (d) 3. The number of vectors in a basis for vector space is known as

degree (b) order (c) dimension (d) zeros 4. Which is the standard basis vector of ?

(b) (c) (0,0,0) (d) 5. Which of the following is in reduced row echelon form?

(b) (c) (d) 6. For homogeneous system of equations , if , then the system has

trivial solution (b) non trivial solution (c) infinite no. of solutions

no solution 7. Which is not correct for the matrices?

(b) (c) (d) None of these

07 1. is

negative definite (b) positive definite

indefinite (d) positive semi definite 2. If , then eigen values of are

(b) (c) (d) 3. Which of the following linear transformation is one to one?

(b)

(d) None of these 4. Find the value of ‘ ’ if is solenoidal.

(b) (c) (d) 5. Rank of the identity matrix is

n (b) n-1 (c) n+1 (d) 6. For which value of k, (2,1,3) and (1,7, k) are orthogonal?

(b) (c) 0 (d) 7. Dimension of is

0 (b) 1 (c) 2 (d)

Prove by Wronskian that are Linearly independent.

Is a basis for ?

Check whether the set of all with binary operations and is vector space or not.

Determine whether is a linear transformation.

Solve by Cramer’s rule.

Find a standard basis vector that can be added to the set to produce a basis for .

Find values of for which and are not invertible matrices.

Find rank of the matrix by the determinant method. Solve by Gauss-Jordan elimination.

07

Let . Verify that is an inner product.

Find the directional derivative of in the direction of at .

Use Gram-Schmidt process to transform basis into an orthonormal basis with Euclidean inner product.

03 Verify Caley – Hamilton theorem for .

For linear transformation , , which of the following are in ?

(ii)

Identify the curve.

Find the angle between the surfaces and 0 at the point .

Find the work done in moving a particle in the force field 0 along the straight line from to .

07 Using Green’s theorem evaluate where is the boundary of the triangle whose vertices are