PLX HelpLet V and W be finite-dimensional vector spaces. Show that the mapping
A: L(V,W) -> L(W*,V*), A(T)=T*
is linear and bijective.
A) Where's the work? This isn't an answer service
B) Can you define your notation, in particular what $\displaystyle T^*$ is. Is it $\displaystyle T^*:W^*\to V^*:\varphi\mapsto \varphi\circ T$? If so, I can give you the hint that it's evidently injective and linear (this is for you to answer) and surjetivity follows from a dimension argument (noting that $\displaystyle \dim\text{Hom}\left(V,W\right)=\dim\text{Hom}\left (W^*,V^*\right)$
so you are saying if I want to prove that [Math]\varphi\mapsto \varphi\circ T[/tex] is linear. I have to find something that live in$\displaystyle \varphi\$ and that prove it is linear???
here is what I am trying to do let g,h in $\displaystyle \varphi\ $ and c,d are scalars
so (T*(cg+dh))(v) = (A(T)(cg+dh)(v)=....= (cT*g+dT*h)(v)
Am I get it???
For proving they are bijective I have to show that dimV=dimW.
But how can I find the dim V and dim W???
if it is surjetivity then the dim V>= dim W but how can I find that dimW?
as I know that dim V= dimT(V) + Ker (T) but where is T(v) lives?
Is it live in V or in T this linear map when we apply V then we have W.
I know that surjective is all V apply linear map T is on W.
But for me it is so hard to show that everything is on W.
I have problem on showing that is surjective.
Plx help.