Homomorphisms and isomorphisms

[happyg1]

Homework Statement

Let$$v_1,v_2,...v_n$$ be a basis of V and let $$w_1,w_2,...w_n$$ be any n elements in V. Define T on V by
$$(\lambda_1 v_1+\lambda_2 v_1+...+\lambda_n v_n)T=\lambda_1w_1+...\lambda_n w_n.$$
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 $$v_i$$ 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

 

[matt grime]

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?