Antima Jangid posted an Question
January 02, 2022 • 15:28 pm 30 points
  • IIT JAM
  • Mathematics (MA)

Question no. 65

(Net Dec 2012) 0. let n he a positive integer and V be an (n +l) dimensicnal vector space over R IF 55. Let S:R>R° be given byV>rv for a fixed ER,T #0. Let T :R R be a linear transformation such that B=...is a set of en 5 a basis of V and7:V >V 5S the linear transformation satisfying 7(e)= £ for i=12.and T(e..)=0. Tinearly independent eigenvectors of T. Then (a) The matrix of T wth respect of B is dlagonal (6) The matrix of T - S with respect of B is diagonal (a) Nullity of T is I nnovative Institute of Math em atics, S-10, Mahaveer Nagar, Near Jaip ur Hospital, Jaip ur ontacts: /792988708, 8696149555 IM nstitute for JAM, NET, B.SC., M. Sc. (Ent.), I &1 Grade (C) The matrix of 7 with respect of B is diagonal 59. Iet M, (K) denote the space of all nxn matrices but the matri of (T - S) with respect to B is not with entries in a field K. Fx a non-singular matrix diagonal (d) The matrix of T with respect of B is not necessarily diagonal, but upper triangular A=(4,) EM,(K), and consider the linear map T:M,(K) -> M, (K) given by: T(X) =A(X)

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