anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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Zarkon
  • Zarkon
Where are you stuck
anonymous
  • anonymous
i'm just not sure how to prove them
Zarkon
  • Zarkon
start with \(T_1\) being injective

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Zarkon
  • Zarkon
we need to show that if \(T_1(v)=T_1(w)\) then \(v=w\)
Zarkon
  • Zarkon
right?
anonymous
  • anonymous
right
Zarkon
  • Zarkon
so assume \(T_1(v)=T_1(w)\) let \(z=T_1(v)=T_1(w)\) then \(v=T_2(T_1(v))=T_2(z)=T_2(T_1(w))=w\)
anonymous
  • anonymous
ohhh ok.
Zarkon
  • Zarkon
can you try the surjective part? it is also really short
anonymous
  • anonymous
i'm just not sure how to prove surjective, i know what it means, btu am unsure how to show it withoutnumbers or anything
Zarkon
  • Zarkon
so you need to show that for all \(v\in V\) then there exists a \(w\in V\) such that \(T_2(w)=v\)
Zarkon
  • Zarkon
can you think of a \(w\) that would work?
anonymous
  • anonymous
uhm T(w) ?
Zarkon
  • Zarkon
We know that \(T_2(T_1(v))=v\) correct?
anonymous
  • anonymous
yes! ok so that wouldn't work .. T(v) ?
Zarkon
  • Zarkon
yes...\(w=T_1(v)\)
anonymous
  • anonymous
ohhh ok. so when w=T1(v) T2(w)=v T2(T1(v))=v ... right?
Zarkon
  • Zarkon
correct
Zarkon
  • Zarkon
I would srite it like this... T2(w)=T2(T1(v))=v
Zarkon
  • Zarkon
*write
anonymous
  • anonymous
oh ok, thank you.
Zarkon
  • Zarkon
yw

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