Thanu Dharshani posted an Question
August 23, 2021 • 07:33 am 30 points
  • IIT JAM
  • Mathematics (MA)

An pic.a let t:r->r'be defined by t( , xz, x)= (r-x, x-x,0).if n(t) and r(t) denote the null space and the ránge space of t respectively, then (a) dim n(t) = 2

an pic.a Let T:R->R'be defined by T( , Xz, x)= (r-X, X-x,0).If N(T) and R(T) denote the null space and the ránge space of T respectively, then (a) dim N(T) = 2 c) R(T) =N(T) (b) dim R(T)=2 d) N(T)C R(T) (JAM MA - 2006)

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    Irfan ansari 1 best-answer

    Let T : R3---> R3 be def ined by T (x1, x2, x3) = (x1 - x2, x1 - x2, 0). and N(T) and R(T) denote the null space and the range space of T respectively. Let (x1, x2, x3) ∈ ker T. Then T(x1 x2, x3) = (0, 0, 0) using the definition of T, we get (x1 - x2, x1 - x2, 0) = (0, 0, 0) Implies x1 = x2, x3 is arbitrary. Therefore,ker T= {(x1, x2, x3) : x1, x3 arbitrary} Hence, nullity of T = 2 and Rank of T = 3 - 2 = 1.

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