BSc Sem IV 2022 2023 Mar 2023 MATHEMATICS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 For one subquestion (each 8 marks) from part
   • (a), and from part (b) Det ine on and Prove that Inverse of a linear transformation if exist V,W be a nite dir Spaces and —W bea transformation then pr ve that
2. Q1 Verify Ra: k-Nullity he ransformation as 2)Show that the following vector Spaces are isomorphic by explicitly defining isomorphism R? and where is set of all polynomial of degree<2 3)For a real vector Spaces V,W, linear transformation T:V—Wprove that i)T(-x)=
   • iii) SV
3. Q2 (a) Attempt any one. [each 8Mks] and prove Cauchy-Schwarz Inequality for an inner product space (V,<>)
4. Q2 Prove that the a parallelogram is a rhombus iff diagonals are perpendicular to each
   • (b) Attempt any two. [each 6Mks] 1)In an inner product space V=R? with Find i) ii) distance between(4,2),(-1,3) in this space iii) prove that the vectors 3 (2,1),(-9,8) are orthogonal vectors in this space 2)Find the orthonormal basis corresponding to the basis of {(1,0,3),(2,1,1)} using VCD/ SEM IV - MATHEMATICS II- 75MARKS 3)Let an inner product space Slet then prove ii)
5. Q3 (a) Attempt ar all eiger vectors of A associated with then
   • ii)Prove that ei al matrix A are the diagonal entries of A Al pe 2x i stics | polynomial of A then prove that
   • (b) Attempt any twe = 3)Suppose eigen values of a A are 1,2,3 then prove that exists and also find
6. Q4 Attempt any three. [each 5 Mks] 1)Find the matrix associated with the linear transformation T:R?—>R‘ with respect to the standard bases of where = 2) Prove that as is a linear transformation
7. Q3 Find projection of p(x)=x on q(x)=x+1 p(x)q(x)dx in C[0,1] 4)Find orthogonal complement of the subspace W={(x,y,z) in
8. Q5 Find eigen value of linear transformation given by 6)Find the quadratic form associated with the symmetric matrix

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