Question

Suppose that V is a vector space over R (not necessarily finite dimensional), and that T1 : V −→ V and T2 : V −→ V are linear transformations from V to V with the property that T3 = T2 ◦ T1 is the identity transformation, i.e. that T3(v) = v for all vectors v in V .
(a) Prove that T1 is injective.
(b) Prove that T2 is surjective.

Answer 1

maybe you could use this: talking about linear transf., is talking about matrixes. since T3 = T2 ◦ T1 is identity transformation. It means that T2 is inverse of T1. Matrix only have inverse if its regular. From here shouls folow that any vector in Space Image would have unic corespondence with vector in Space Domain. And since the matrixes we talking about are square, same for the inverse situation. Not sure tough.... I am a bit rusty with algebra

Answer 2

maybe experimentX can add something better, :)

Answer 3

seriously can't think a word ... except T2 is inverse of T1

Answer 4

but i guess it should go more or less this way

Answer 5

well, linear transformation applies not only to matrices .. I can apply to two sets

Answer 6

haven't done algebra for pretty long ... and wasn't good at it though

Answer 7

I mean that matrix sapce is dual to linear transf. space. So to prove something in one you can use the other..

Answer 8

:), me either

Answer 9

I know this ... if a function is bijective, then it's inverse transformation exists

Answer 10

identity transformation and composition of two transformation, I can't think of anything except bijective realation

Answer 11

T2(T1(v)) = v
=> T1(v) = T2'(v)
=> T1 = T2'
bijective relation => injective and surjective

Answer 12

might not be ... just a hunch though.