Question

i have question related to transformation
prove these few statements are equivalent
a) T is non singular
b) if T(A)=T(B) then A+B
c) N(T)={0}
d) n(t)=0
e) r(t)=n
f) if {a1,a2,...an} is basis for Vn then {Ta1,Ta2,...,Tan} is basis for R(T)