MTE-02 Solved Assignment 2019 Linear Algebra (English Medium)

Description

Course Code: MTE-02

Assignment Code: MTE-02/TMA/2019

Maximum Marks: 100

1. a) Which of the following are binary operations onC? Justify your answer.
   i)The operation5defined byx5y=|xy|.
   ii)The operation4defined byx4y=xywhere the bar denotes complex conjugation.
   Also, for those operations which are binary operations, check whether they are associative and commutative.
   b) Find the vector equation of the plane determined by the points(1,1,−1),(1,1,1)and(0,1,1). Also find the point of intersection of the liner= (1+3t)i+(2−t)j+(1+t)kand the plane .
   c) Check whether the vectorsi+j+k√3,i−j√2,i−2j+k√6are orthonormal.
2. a) Check whether the set of vectorsv1= (1,1,0,1),v2= (1,0,2,1),v3= (−1,1,−3,−2)∈R4are linearly indpendent. If they are dependent, findα1,α2andα3∈R, not all zero, such thatα1v1+α2v2+α3v3=0.
   b) Which of the following are subspaces ofR3? Justify your answer.i)S={(x,y,z)∈R3∣∣x+y=z}ii)S={(x,y,z)∈R3∣∣2x=3yz}
3. LetP(e)={p(x)∈R[x]|p(x) =p(−x)}P(o)={p(x)∈R[x]|p(x) =−p(−x)}
   a) Check thatP(e)andP(o)are subspace ofR[x].
   b) Show thatP(e)={∑iaixi∈R[x]∣∣∣∣∣ai=0 ifiis odd.}P(o)={∑iaixi∈R[x]∣∣∣∣∣ai=0 ifiis
   Deduce thatP(o)∩P(e)={0}
   c) Checkp(x)+p(−x)∈P(e)for everyp(x)∈R(x). Check that the mapψ:R[x]→P(e)given byψ(p(x)) =p(x)+p(−x)2is a linear map. Further, check thatψ2=ψ. Determine the kernel ofψ
4. a) LetT:R3→R3be defined by T(x1,x2,x3) = (x1+x3,x3,x2−x3).IsTinvertible? If yes, find a rule forT−1like the one which definesT. If T is not invertible, check whether T satisfies Cayley-Hamiltion theorem.
   b) Find the inverse of the matrix
   2 1 0
   0 1 1
   1 1 0
   using row reduction.
5. a) Check whether the following system of equations has a solution.
   4x+2y+8z+6w=4
   2x+2y+2z+2w=0
   x+3z+2w=3
   b) Define T:R3→R3by
   T(x,y,x) = (−x,x−y,3x+2y+z).
   Check whether T satisfies the polynomial(x−1)(x+1)2. Find the minimal polynomial of T.
6. a) Let T:P2→P2be defined by
   T(a+bx+cx2) =b+2cx+(a−b)x2.
   Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B1={x2,x2+x,x2+x+1} and B2={1,x,x2}. Find the kernel of T.
   b) Consider the basis e1= (1,1,−1), e2= (−1,1,1) and e3= (1,−1,1) of R3 over R. Find the dual basis of {e1,e2,e3}.
7. a) Check whether the matrices A and B are diagonalisable. Diagonalise those matrices which are diagonalisable.
   i) A=
   1 00
   1 2−3
   1 1−2
   ii) B=
   −2−4−1
   3   5  1
   1   1  2
   b) Find inverse of the matrix B in part a) of the question by using Cayley-Hamiltion theorem.
   c) Find the inverse of the matrix A in part a) of the question by finding its adjoint.
8. a) Let P3 be the inner product space of polynomials of degree at most 3 over R with respect to the inner product〈f,g〉=∫10f(x)g(x)dx. Apply the Gram-Schmidt orthogonalisation process to find an orthonormal basis for the subspace of P3 generated by the vectors
   {1−2x,2x+6×2,−3×2+4×3}.
   b) Consider the linear operator T:C3→C3, defined by
   T(z1,z2,z3) = (z1+iz2,iz1−2z2+iz3,−iz2+z3).
   i) ComputeT∗and check whether T is self-adjoint.
   ii) Check whether T is unitary.
   c) Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases B1={(1,0,0),(0,1,0),(0,0,1)}, B2={(1,0,0),(0,1,2),(0,2,1)}. If Q(X) =x21−2x1x2+4x2x3+x22+x23, find the representation of Q in terms of (y1,y2,y3).
9. Find the orthogonal canonical reduction of the quadratic form−x2+y2+z2−4xy−4xz.Also, find its principal axes.
10. Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
    i) R2 has infinitely many non-zero, proper vector subspaces.
    ii) If T:V→W is a one-one linear transformation between two finite dimensional vector spaces V and W then T is invertible.
    iii) If Ak=0 for a square matrix A, then all the eigenvalues of A are zero.
    iv) Every unitary operator is invertible.
    v) Every system of homogeneous linear equations has a non-zero solution.