BSc Mathematics SEM II 2021 2022 2022 ALGEBRA II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Tick mark the correct option (2 mks each)
   • 1) An elementary matrix of order n is matrix
   • A) obtained by applying only one row operation to identity matrix of order n
   • B) obtained by applying only two row operation to identity matrix of order n
   • C) obtained by applying only three row operation to identity matrix of order n
   • D) obtained by applying only four row operation to identity matrix of order n
   • 2)A square matrix is invertible iff
   • A) it can be written as product of nonzero matrices
   • B) it can be written as product of nonzero and some zero matrices
   • C) it can be written as product of elementary matrices
   • D) it. can be written as product of diagonal matrices
   • 3)Row space of matrix [A]mxn is
   • A) subspace of R" generated by row vectors of A ;
   • B) subspace of generated by columnvectors of A
   • C)subspace of generated by row vectors of A
   • D)subspace of generated by column vectors of A
   • 4)The rank of null matrix is
   • 5)State whether the following vectors are .(1,2,3),(2,-2,0)
   • C) both A and B D) neither A nor B
   • 6) A matrix unit is a matrix with dimensions
   • A) but with elements
   • B) but without actual elements
   • C) but all elements are zero = D) but all elements are one
   • A) No elementary matrix B) Every elementary matrix C)some of an elementary matrix D)any
   • 8) Let AX =B be a system of n linear equations in n unknowns.IF IAI =0 then
   • A) system has unique solution
   • B) System has infinitely many solutions
   • C)System has no solutions
   • D)any one of B or C
   • 9)In Gaussian Elimination method,to solve AX=D,the augmented matrix [A/D] is reduced to
   • A) identity matrix
   • B) null matrix
   • C) an upper triangular form
   • 10) Which of the following is false?
   • A) The homogeneous system of equations has a unique solution
   • B)The homogeneous system of equations has infinitely manysolution
   • C)The homogeneous system of equations has no solution
   • D)The homogeneous system of equations has zero solution
   • 11) There are ----- types of elementary matrix
   • A) three types of an elementary matrix
   • B)AN infinite types of an elementary matrix
   • C) only one type of an elementary matrix
   • D)as per order of the matrix
   • 12) Let A be any nonsingular square matrix,B is a matrix of same order such that
   • A)AB=I B)BA=I C)AB=BA=I D)AB=BA=O where O is null matrix of same order
   • 13)Matrix is a vector space over
   • 14)Additive Identity in a vector space
   • A)Not uniqueB)Unique C)1D)Either 1 or 0
   • 15)Let V be a vector space and W be a non-empty subset of V. Then W is a subspace of V if and
   • A)a.x W B)a.x—bye V C)axtbyeW
   • 16)Dimension of Vector space is,
   • 17)Which of the following is not a subspace or
   • D)Planes passing through origin in
   • 18)Which of the following in a vector space?
   • A)S = {(x, y,x ye R}
   • C)S = {(x, y, 2)/ x, y R} on
   • 19)The of two subspaces of a vector space V is always a subspace of V
   • 20)Union of two subspaces of a vector space V is of V if and only if
   • A)They are disjoint B)One is a subset of other
   • C)Both are subsets of each other of finitely many subspaces of a vector space V is also a subspace of V
   • A)Addition
   • D)Union as well as intersection
   • 22)Which of the following set of matrices of real matrices of order 2 forms a subspace of M, skew symmetric matricesB)AII invertible matrices C)AIl non-invertible matricesD)All matrices A such that A*=A
   • 23)Let V be a vector space, and let A & B be two subspaces of V such that either
   • A)A U B is not a subspace of V B)A B is not a subspace of U Bisa
   • D)A U B may or may not be a subspace of V
   • 24)Inverse of a linear
   • A) need not be a linear transformation
   • B)is again a linear transformation
   • C) is not always linear transformation
   • D)may not exist always
   • 25)Find the linear transformation T from R’such that T(1,0)=(4,3,-1) and T(0,1) =(-5,6,1))Find the linear transformation T from R? to that T(1,0)=(4,3,-1) and T(0,1)
2. Q2 Attempt any three of the following (5 MKS each)
   • 1) Prove that a system of linear equations has either no solution or unique solution or
   • 2) Prove that for A,B matrices of order mxn then A,B are row equivalent iff there exists a invertible matrix P such that B=PA
   • 3)Prove that A is invertible iff A can be expressed as product of elementary matrices the following system of linear equations using Gauss elimination method
3. Q3 Attempt any three of the following (5 MKS each)
   • 1)Prove that P[x] denote set of all polynomials with real coefficiens then it is a vector space over R with respect to usual addition and scalar multiplication of polynomials
   • 2) Check whether A is a subspace of R? where
   • 3)Let (V,+,*) be a real vector two subspaces of V then prove that intersection of W, and W, is also a subspace of V
   • 4)Prove that S={(1,0),(0,1)} is a basis of R?
4. Q4 Attempt any three of the following (5 MKS each)
   • 1)Define a Kernel of a linear transformation and find Kernel of T for as
   • 2)Let V be a finite dimensional real vector be a linear transformation then KerT
   • 3)Prove that a linear transformation is invertible iff T is an isomorphism
   • 4)If is one one linear transformation then what should be the relation between
5. Q5 Attempt any one of the following (5 MKS each)
   • 1) Show that the matrices A,B are row equivalent by producing a sequence of elementary row Operations that produces B from A
   • 2)Let (V,+,*) be a real vector space then prove that additive identity is unique
   • 3) Check whether the following is a linear transformation or not

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