Proove that linear transformation is one one but not onto.

PFA.

is there is any relationship between "an" and "n" or be take any arbitrary relationship which is one-one but not onto? if there is relationship between "an" and "n" we will prove ...

T(An) = (0,0,...) implies (0,a1, a2,....) = (0,0,0,...) implies a1= a2=.....= 0 implies An =(0,0,0, ..) implies Kernel T = {(0,0,0,...)} . Hence, T is one one