Let V - finite dimensional vector space over field, F
T,S be linear transformation from V into/onto V
I be the identity tranformation
Q1: m(x) is minimal polynomial of T
p(x) is characteristic polynomial of T
if q(x) is an irreducible polynomial over F and q(x)|p(x), prove that q(x)|m(x)
as well.
Q2: If ST=I => TS = I
(basically I want to prove if a transformation is right or left invertible, it is invertible)
Q3: T^n = 0 => T=0. I am not sure of this one, but couldn't find the reason/counter-example
Any pointers would be welcome. I am just picking up the subject. So these might really be elementary question but will appreciate any help here.
Thanks
Hint: in general if then
is 1-1 because it's onto. we also have for all because thus because is 1-1.
Q2: If ST=I => TS = I
(basically I want to prove if a transformation is right or left invertible, it is invertible)
you forgot to mention what is! whatever it is, the implication is false. for example, on define then but
Q3: T^n = 0 => T=0. I am not sure of this one, but couldn't find the reason/counter-example
Any pointers would be welcome. I am just picking up the subject. So these might really be elementary question but will appreciate any help here.
Thanks
Thanks very much. I'll try your hint to Q1.
Let T:V->V be a linear transformation. Am I correct in saying:
T is onto IFF T is one-one
Proof: Let v1,v2, v3...,vn be basis of V
T is onto => T(v1), T(v2), ... T(vn) span V hence is a basis of V. So if T(v) = 0 => v = 0. Which further => T is one-one
T is one-one. T(v1), T(v2)....T(vn) is basis for range of T which is a equal to/subspace of V. But any subspace of V with n basis = V itself hence T is onto.
Is my reasoning correct?