BSc Mathematics SEM III 2017 2018 2018 Mathematics Paper II Question Paper - Mumbai University | munotes

Questions asked in this paper

• 2. Figures to the right indicate marks to respective parts

1. Q1 Choose correct alternative in each of the following: Let A= ( ) and 5 the standard matrix for the transformation T defined by T(x)=A(Bx) is ll. The range of the linear transformation from to R? given by the 20 marks
   • (a) R? (b) Aline through the origin
   • (c) Aline through the (d) A line through origin having points (2,3) and (-6,-9) slope 2/3 Let T be the linear transformation that T(1, 0) = (4, 3) and lv. Area of the parallelogram spanned by vectors (—1,2) and (3,4) is
   • (c) 2 (d) None of the above
   • (c) (0 1 O (d) None of the above
   • vi. Which of the following is false for an invertible n x n matrix A?
   • (c) det = (det A)? (d) None of the above If , lige Ma(R) then is vill. of the following set forms group under the given binary 1x. The order of the group S,,is
   • (c) None of these
   • x. The identity element of the group G = { /a } under multiplication of 2x 2 matrices is
   • (c) ( D (d) None of these
2. Q2 a) Attempt any ONE question from the following: 8 marks
3. Q1 Let. V,W be vector spaces over R and be a linear transformation and if V is finite dimensional then show that
   • ii. Show that the following are equivalent for a linear map T:
4. Q2 Ker T = {0}
   • b) Attempt any TWO questions from the following: 12
   • i. Let T:IR? > R*be defined as T(x) = Ax where A is the matrix of T with respect to standard bases {e1, e2} on both sides and A= 1): What is the matrix of T with respect to changed bases e2} on both sides? Let > Rbe defined as T(x) = Ax where A= (2 i) Determine rank T, nullity of T and hence verify the rank — ili. Let V be the vector space of real polynomials in the variable x and let D? V-V defined as = then find ker Also find its dimension Let A= the matrix of linear transformation T:R? > defined as T(x) = Ax then show that T is invertible and find formula for
5. Q3 a) Attempt any ONE question from the following: 8 marks
   • i. Let @:R* x Rbe a bilinear function such that standard unit vectors of Prove that = det(A?, A”) for any column vectors A? R? Prove that the row rank and the column rank of anm x n matrix A are equal
   • b) Attempt any TWO questions from the following: 12
   • i. Express A as product of elementary matrices where Define row rank ,column rank of A Find rank of A -(3 4 -1 2) If At, A” are n linearly dependent column vectors then prove that = 0
   • iv. Solve the following system of linear equations using
6. Q4 a) Attempt any ONE question from the following: 8 marks
   • i. Define Group. Prove that the set of residue classes modulo n, Zn 18 a group under addition modulo n
   • i. Let G bea group. For a,b,x G Prove that
7. Q11 o(ab) = o(ba)
   • b) Attempt any TWO questions from the following: 12
8. Q1 Let G be a group and H be non-empty subset of G. Prove that H is a subgroup of G iff H,Va,b H il. Consider D3 = ba, where a? = = e and ab = ba’. Prove that D3 is a group under composition of functions. Is it abelian? ili. Construct composition tables of U(8) and U(10) groups. Are they examples of Klein-4 group? Justify your answer lv. Let G group. For any a,b G, prove that
9. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) Show that linear map T:R? > given by T(x,y) = (2x + y,x + y,x) represents one-one linear transformation and linear map S:R? > R? given by S(x,y,z) = (x + z,x + y) represents onto linear
   • b) Show that > R* defined as T(x,y) =(x+2y, x — y) is a linear
   • c) Determine the value of k for which the following system of linear equations has no solution: Using adjoint of a matrix, find 5
   • e) Construct composition table of under multiplication modulo 5. Also find order of all elements of Zé
   • f) Show that H= {1,, (13)0(24), is a subgroup of

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