Page 1 :
Total No. of Questions Pe [Total No. of Printed pages, BA/B.Sc. V - Semester Examination, CBS-Vs/5 (S), 220559, MATHEMATICS, , Course No. : UMTTE - 503, , - Time Allowed : 2% Hours Maximum Marks : 80, , SECTION-A, , Note: Attempt all questions from this section. Each question carries 4, marks, Y, , 1. Show that intersection of two subspaces of a vector space, isa subspace. Is the result true for union? Ifnot, why? = (2,1,1), 2. Show that every quotient space is homomorphic image of the, v-space. (4), 3. Show thata linear transformation f :U (F)3V(F ) is 1-1 iff, kerf={0}. (2,2), 4. . Show that eigen values ofa hermitian matrix are real. (4), , 5. Define linearly independent and linearly dependent set of vectors,, Show that if, y, z are L.J, then so are x+y, ytzandztx. (4)

Page 2 :
(2) CBS-Vs/5(S)-220559, SECTION-B, Note: Attempt all questions from this section. Each question carries 8, , marks,, , 6. Show that a subset 4 = {%)5X),...,} of non-zero vectors, is linearly dependent ina v-space V(F) iff some vector, , %, €4.2<kXn is linear combination of its preceding vectors., , (4,4), 7. IfU(F)and Vv (F) are fd vector spaces and, f:U(F)> V(F) isa linear transformation, then, Rank f+Nullity f=dim.U. . (4,4), 8. State and prove Cayley-Hamilton theorem. (1,7), , SECTION-C, , Note: Attempt any two questions from this section. Each question, carries 18 marks., , 9. IfAand Bare subsets of a v-space V(F), then prove that, i) L(A) is the smallest subspace of V(F) containing A., , i) Ac B= L(A) cL(B), , i) L(AUB)=L(A)+L(B), yt iv) Aisasubspace of V(F) iffL(A)=A, , y L(L(A))=4(4). (4,4,4,4,2)

Page 3 :
10. i), , ii), ll. 3), ii), 2. 9, i, , (3) CBS-Vs/5(S)-220559, , If A and B are. f.d subspaces of a f.d v-space VF), then, prove that dim., , (A+B) =dim. A+dim.B-dim.(A B)., , Find Rank of the matrix (12,6), , ARN, oon:, , 3, 6, 1, , Show that every finite dimensional vector space ve ), , ‘has a basis., , Show that Sey finite dimensional vector space is, isomorphic to its dual space. (9,9), Show that the set V ofall mxn matrices over a field F, is a vector space under usual addition and scalar, multiplication., , State and Prove fundamental een of v-space, , homomorphism. (9,9)