12-18-2014, 09:29 PM

FIRST SEMESTER M.Sc. (MATHEMATICS)DEGREE EXAMINATION (CUCSS 2010)

Model Question Paper

MT1C02 Linear Algebra

Time: 3 Hours Max. Weightage: 36

Part A( Short Answer Type Questions)

Answer all the questions

(Each Questions has weightage one)

1. Give an example of a vector space and prove your claim.

2. Prove that the only subspaces of R 1 are R 1 and the zero subspace.

3. Construct two bases for R

3 which has no common elements.

4. Prove that if two vectors are linearly dependent, one of them is a scalar multiple of the other.

5. Define Differentiation transformation and find its null space.

6. Let T and U are two linear transformations . Prove or disprove that T U = UT.

7. Show that F m×n is isomorphic to Fmn.

8. Give an example of a nonsingular linear operator and prove your claim .

9. Define hyperspace in a vector space and give an example.

10. Give an example of a linear operator on a vector space which has no characteristic value.

11. Let V be a finite dimensional vector space. What is the minimal polynomial for the identity operator on V .

12. If E1 and E2 are projections onto independent subspaces, then prove or disprove that E1+E2 is a projection.

13. Give an example of an inner product space and prove your claim.

14. Let T be a linear operator on the n-dimensional vector space V , and suppose that T has n distinct characteristic values. Prove that T is diagonalizable.

Part B ( Paragraph Type Questions)

Answer any Seven questions

(Each question has weightage two)

15. Let V be a vector space over the field F. Show that the intersection of any collection of subspaces of V is a subspace of V .

16. Let W be a subspace of a finite-dimensional vector space V , show that every linearly independent subset of W is finite and is part of a basis for W.

17. Let V and W be vector spaces over the field F and let T be a linear transformation from V into W. If T is invertible , then prove that the inverse function T−1 is a linear transformation from W onto V .

18. State and prove Rank-Nullity theorem.

19. Let T be a linear transformation from V into W. Then prove that T is non singular iff T carries each linearly independent subset of V onto a linearly independent subset of W.

20. Let T be the linear operator on R 2 defined by T(x1, x2) = (−x2, x1).

a) What is the matrix of T in the standard ordered basis for R2?

b) What is the matrix of T in the ordered basis C = {α1, α2}, where α1 = (1, 2) and α2 = (1, −1) ?

21. Let A be any m × n matrix over the field F. Then prove that the row rank of A is equal to the column rank of A.

22. Let T be a linear operator on an n- dimensional vector space V . Prove that the characteristic and minimal polynomials for T have the same roots, except for multiplicities.

23. If W is an invariant subspace for T, then prove that W is invariant under every polynomial in T. Also prove that the conductor S(α; W) is an ideal in the polynomial algebra F[x] for each αin V .

24. Show that an orthogonal set of non-zero vectors is linearly independent.

Part C (Essay Type Questions )

Answer Any Two Questions

(Each Question has weightage Four )

25. (a) Let V be a finite dimensional vector space and let T be a linear operator on V . Suppose that rank (T2) = rank (T). Prove that the range of T and the null space of T are disjoint, i.e, have only the zero vector in common.

(b) Let V and W be finite dimensional vector spaces over the field F such that dim V = dim W. If T is a linear transformation from V into W, then prove that the following conditions are equivalent:

(i) T is invertible.

(ii) T is non-singular.

(iii) T is onto, that is, the range of T is W.

26. (a) Define transpose of a linear transformation.(b) Let V and W be vector spaces over the field F, and let T be alinear transformation from V intoW. Show that the null space of Ttis the annihilator of the range of T. If V and W are finite

dimensional,then prove that(i) rank (Tt) = rank (T)(ii) The range of Ttis the annihilator of the null space of T.

27. State and prove Cayley-Hamilton Theorem.

28. (a) State and prove Gram-Schmidt orthogonalization process.

(b) Let V be vector space and (|) an inner product on V . Show that if (α|β) = 0 for all β ∈ V , then α = 0 .