BSc Mathematics SEM II 2018 19 May 2018-19 MATHEMATICS PAPER II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 All questions are compulsory
2. Q2 Figures to the right indicate marks for respective parts
3. Q3 Use of Calculator is not allowed
4. Q1 Choose correct alternative in each of the following: 20 marks
5. Q1 A linear system of equations may have
   • (a) No solution (b) Unique solution
   • (c) infinitely many solutions (d) Any of (a), (b), (c) li. A diagonal matrix D = is invertible if and only if iil. Let A= Then A" (where n is positive integer) is
   • (a) for all n (b) A
   • (c) I, if nis even and A if n is odd (d) None of these Iv. Which of the following is a subspace of over R?
   • (c) = None of the above S 1), (0, 1, 0)} then the linear span of S is
   • (c) XY-plane R Vi. The singleton set {0} in any real vector space V is
   • (c) a basis of the empty set (d) none of the above Which of the following set is a generating set of R*? The dimension of the vector space of all real matrices of order 2 x 3 is
   • (c) 13 (d) None of these Ix. Which of the following is a linear transformation? by (b) T:R? R*defined by
   • (c) T: R? R? defined by (d) None of the above If for a linear transformation T: Rank T = 4 then the nullity of T is
   • (c) (d) None of these
6. Q2 a) Attempt any ONE question from the following: 8 marks
7. Q1 Prove that the homogeneous system a,x + byy = + boy = 0 has a non-trivial solution if and only if = 0 li. Define trace of a square matrix A (Tr(A)) of order n. Show that if A and B are square matrices of then Tr(A) = and Tr(A + B) = Tr(A) +
   • b) Attempt any TWO questions from the following: 12
8. Q1 For any square matrix A of order n, prove that
   • (p) A+A’ is symmetric (q) is skew symmetric li. Show. that the following system of equations has non-trivial solution for all
   • ii. If A is a square matrix then prove that
   • (p) If A? = O then I-A is invertible. (q) If = O then A +I is invertible Iv. Show that the following system of linear equations is consistent using Gauss
   • a) Attempt any ONE question from the following: 8
   • i. Let V be vector space over R and V = . be linear independent For u V, prove that S U is linear dependent if and only if u where L(S) is linear span of S li. Verify all properties of scalar multiplication of the vector space = {| Ao lai; R}, where the operation of matrix addition (+) and scalar multiplication (.) are defined as usual
   • b) Attempt any TWO questions from the following: 12
9. Q1 Show that in any vector space the subset of a linear independent set is linear li. Prove that the set W = {A = BA}is a subspace of where Bis a fixed matrix in
   • iii. 2 3 3] sO Express as a linear combination of F ; 4 | 0 Iv. Let u,v and w be linear independent vectors in real vector space V, show
10. Q4 a) Attempt any ONE question from the following: 8 marks
11. Q1 Prove that in a vector space of dimension n any set containing n + | vectors is linear dependent li. Define kernel and image of a linear transformation. If T:V V’ is a linear transformation, prove that Im(T) is a subspace of V’
    • b) Attempt any TWO questions from the following: 12
12. Q1 Prove that the vectors (1,1,0), (1,2,3) and (2,-1,5) form a basis for R?
    • ii. Extend S = {(1,0,—1)} to a basis of Let : V Wbe two linear transformations, Prove that + ~ W is also.a linear transformation
13. Q4 State Rank-Nullity theorem and verify it for the linear transformation
14. Q5 Attempt any FOUR questions from the following: 20 marks
    • a) Find parametric representation for the plane passing through point P = (1,1,1),
    • b) Reduce the matrix E 1 to it echelon form Let S = {(1,0), (0,1)} and T = {(1,1), (1,-1)}. Prove that L(S) = L(T)
    • d) Show that any two bases of a finite dimensional vector space have the same number of vectors
    • e) Find a linear map T: R? > R* such that T(1,0,1) = (2,3), T(0,1,0) = (0,1),
    • f) Find the matrix of the linear transformation T : R? R? defined by T(x, y) = (x — 2y,3x + 4y,x — 6y) bases of and

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