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 .

Prove that T1 is injective.
Prove that T2 is surjective.

Answer 1

Where are you stuck

Answer 2

i'm just not sure how to prove them

Answer 3

start with \(T_1\) being injective

Answer 4

we need to show that if \(T_1(v)=T_1(w)\) then \(v=w\)

Answer 5

right?

Answer 6

right

Answer 7

so assume \(T_1(v)=T_1(w)\)

let \(z=T_1(v)=T_1(w)\)

then \(v=T_2(T_1(v))=T_2(z)=T_2(T_1(w))=w\)

Answer 8

ohhh ok.

Answer 9

can you try the surjective part? it is also really short

Answer 10

i'm just not sure how to prove surjective, i know what it means, btu am unsure how to show it withoutnumbers or anything

Answer 11

so you need to show that for all \(v\in V\) then there exists a \(w\in V\) such that \(T_2(w)=v\)

Answer 12

can you think of a \(w\) that would work?

Answer 13

uhm T(w) ?

Answer 14

We know that \(T_2(T_1(v))=v\) correct?

Answer 15

yes! ok so that wouldn't work ..
T(v) ?

Answer 16

yes...\(w=T_1(v)\)

Answer 17

ohhh ok. so when w=T1(v)
T2(w)=v
T2(T1(v))=v ... right?

Answer 18

correct

Answer 19

I would srite it like this...

T2(w)=T2(T1(v))=v

Answer 20

*write

Answer 21

oh ok, thank you.

Answer 22

yw