Need some help with linear transformations.

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Need some help with linear transformations.

Mathematics
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The equation I'm given is: \[T(a _{0}+a _{1} x+a _{2} x^2)=a _{1}+2a _{2} x\] I'm supposed to determine if it is a linear transformation from P2->P2, but I don't think it is, I think it is from a P2-> P1, I'm not sure how to prove it?
It is differentiation, which is a linear transform. \[T(a_0+a_1x+a_2x^2)=a_1+a_2x+0x^2\]
I don't know what a differentiation is? The part that confuses me is the \[2a _{2}x\] They are mixed?

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oh sorry, i forgot to put in the 2.
T(a0+a1x+a2x2)=a1+2a2x+0x2
Right, but the A2 is with the X not the X^2
So the right side is to the first degree so doesn't that make it a P1?
Can anyone out there help explain this to me?
bummer no one can explain, thanks anyway :-)

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