[SOLVED] Isomorphisms question

Okay, so I want to show that the transformation
T(a+bx+cx^2)=a+2bx+4cx^2 from P2 to P2 (from a second degree polynomial to a second degree polynomial) is an isomorphism.

I know I have to show that ker(T)=0, but I'm not sure how to do that for this particular problem. Any ideas?

Quote:

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Originally Posted by joej

Okay, so I want to show that the transformation
T(a+bx+cx^2)=a+2bx+4cx^2 from P2 to P2 (from a second degree polynomial to a second degree polynomial) is an isomorphism.

I know I have to show that ker(T)=0, but I'm not sure how to do that for this particular problem. Any ideas?

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Let p(x)=a+bx+cx^2, then: T(p)(x)=a+2bx+4cx^2, and if p in ker(T), we
have:

T(p)(x)=a+2bx+4cx^2 = 0

in which case we have a=0, b=0, c=0, ie p is the zero polynomial, ie ker(T)={0}.

RonL