BSc Mathematics SEM III ATKT MATHEMATICS PAPER II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Choose correct alternative in each of the following 20 marks
   • i. isa linear transformation if Vu,v R,
   • (c) T(au+ None of the above
   • ii. —V isa linear transformation then
   • (d) All of the above
   • iii. Which of the following is a linear transformation from R? to
   • (c) + y, X- y) (d). All the above
   • iv. _(4 (0 3 IfA and EA , then 1S by
   • v. Which one of the following is NOT TRUE
   • vi. where e;, are standard basis elements of R? is
   • vii. Let A M,(R) be an invertible matrix then det(Adj A) is
   • (c) (d) None of these Let V be a finite dimensional inner product space and W be a subspace of V Then is equal to
   • ix. For x = and y = which of the following is not an inner
   • x. If {v,, an orthonormal basis for R*with Euclidean inner product, then for :
   • (c) O (d) None of these
2. Q2 Attempt any ONE question from the following: 8 marks
   • a) i. State and prove the Rank-Nullity Theorem
   • ii. Let A M,,(R). Prove that the system AX=B of n non-homogenous linear equations in n unknowns has a unique solution if and only if rank(A) =
3. Q2 Attempt any TWO questions from the following: 12 marks
   • b) 1. Show that F is non-singular where F : given by
   • ii. given by (x+ 2z). Find the basis for Ker T and Nullity T Show that a n-dimensional real vector space is isomorphic to
   • iv. Test for consistency and if consistent solve the system:
4. Q3 Attempt any ONE question from the following: 8 marks
   • a) i. Prove that Ais invertible if and only if columns of A are linearly independent. Hence, prove that if det A = 0 then columns of A are linearly dependent
   • ii. Let IR”. Show that
5. Q2 =
6. Q3 Attempt any TWO questions from the following: 12 marks
   • b) i. Let A show that = detA , where is the
   • ii. Solve the following system of linear equations using Cramer’s rule
   • iii. For A,B M,,(R), if A is invertible show that
   • iv. Define adjoint of a matrix. Find for A = ( 2 using adjoint
7. Q4 Attempt any ONE question from the following: 8 marks
   • a) i. Define inner product and inner product space over R. Show that , )) is an inner product space over R where
   • ii. Define orthogonal and orthonormal sets. Let be an orthonormal basis of an inner product space V Let xX = + + Then prove the following:
   • (q) Il x
8. Q4 Attempt any TWO questions from the following: 12 marks
   • b) i. Define angle between two vectors in a real inner product space. Find angle between A = ( 1 and B = ( 1 with respect to the inner product (A, B) = tr(AB‘) on M,(R)
   • ii. Prove that an orthogonal set in a real inner product space V is linearly
   • iii. Let W be a subspace of a real inner product space V. Define the orthogonal complement of W. Show that W+ is a subspace of V
   • iv. Apply Gram-Schmidt process to obtain orthogonal set corresponding to {(0,1,1), (4, —1,0), (2,0,1)} in R? with dot product
9. Q5 Attempt any FOUR questions from the following: 20 marks
   • a) Prove that if —V’is a linear transformation then T is injective if and only
   • b) Find the rank of A=
   • c) Use the following expression of determinant to find the determinant of the 1 3
   • d) (1) Use determinant to check whether the homogeneous system 1 -6 1 = | 0 | has non-trivial solution. State the result used (II) Use determinant to find area of the parallelogram spanned by vectors x = (5,6) and y = (2,5). State the result used
   • e) bean inner product space and u, v V. Let a, b be nonzero elements of +b. Prove that + bv||=||bu + iff =
   • f) Find distance between f(x) = and g(x) = sinx in using

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