If VVV and WWW are vector spaces, a transformation T:V→WT: V \to WT:V→W is linear if:
T(u+v)=T(u)+T(v)T(u+v) = T(u) + T(v)T(u+v)=T(u)+T(v) and T(cu)=cT(u)T(cu) = cT(u)T(cu)=cT(u)
T(uv)=T(u)T(v)T(uv) = T(u)T(v)T(uv)=T(u)T(v)
T(0)=1T(0) = 1T(0)=1
TTT is a bijection.