Question

Find an example of a vector space V, and a linear transformation T : V + V such that R(T) = ker(T). Your vector space V must have dimension > 2. You may find it helpful to let V be a euclidean space and T a matrix transformation,

Answer 1

det v=1R4 be a vector Space overir. linear transfor- and. Tivov be a -mation is given by T(mixes, as, Kw = (Mo, xu, 0, 0). xet xa Then (ab.e,d) € kert T(x)=(0,0,0,0) of TCabed] =(0,0,0,0) or Ced, o, o) = (0,0,0,0) >> ezo d20. (a,b,0,0) 2 a (10,0,0)+b(0.6oo) Therefore basis for kert is {(1,0,0,0), (0.600)} x= carb.c,d) Now the Standard basis of IR4 is B=} e,=(1,0,0,0), 2(0.600), e32 (0.0.10), ey =(0. B-{ -(0,0,0.13 (C) = (0, 0, 0, 0) Coro, o Tce): 0,0) T(43) C1,0,0,0) T(44)= (0,100) 2

! 2 Im (T) = R(1) Spam (0,0,0,0), (2,0,0,0), (0100) Spank (1,000), (0,1,0,0)]. Since (6.0.0.0) and (0.100) are linearly independent rector then basis for RCD) is ((1.0.0.0), (0.1.0.0] and basis Since, for basis for KercT) RCT) are same so ker(t)=RCT). And M Atrin representation of tis [т], 0 so o o o 0 0 0 0