Homomorphisms and isomorphisms

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Homework Statement



Let[tex]v_1,v_2,...v_n[/tex] be a basis of V and let [tex]w_1,w_2,...w_n[/tex] be any n elements in V. Define T on V by
[tex](\lambda_1 v_1+\lambda_2 v_1+...+\lambda_n v_n)T=\lambda_1w_1+....\lambda_n w_n.[/tex]
a)Show that R is a homomorphism of V into itself.
b)When is T an isomorphism?

Homework Equations


An isomorphism is a 1-1 mapping.


The Attempt at a Solution


I think I have part a) done, although It makes more sense to me to prove that T is a Hom, not R...not sure where the R came from.

For part b, I am thinking that since the [tex]v_i[/tex] are a basis that the Iso would be true if the w's were a basis also?
IK know that an iso occurs when the Kernel is zero..but I'm still kinda confused here.
Any hints or advice will be appreciated.
Thanks,
CC
 

Answers and Replies

  • #2
matt grime
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Obviously, in order for T to be an iso, w_i must be a basis - the image lies in the span of the w_i, and that can only be all of V if they are a basis. That says T an iso implies w_i a basis. What about the converse?

Now, suppose that v is sent to zero by T, i.e. vT=0. Then what does that say about the linear combination of the w_i that v is sent to?
 
Last edited:

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