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University of Calicut Model Question Paper M.Sc.Mathematics(CUCSS)-ISEM-LinearAlgebra

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 .

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