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Can someone help me with the following exercise?

A basis is given by $(1,x+1,(x+1)^2)$. There is a unique linear trans- formation T sending the basis $(1,x,x^2)$ to the basis $(1,x+1,(x+1)^2)$. Express the matrix of T relative to the basis $(1,x,x^2)$, and then also relative to the basis $(1,x+1,(x+1)^2)$. What is the relationship between these two matrices?

Thank you in advance

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up vote 2 down vote accepted

Hint: The columns of the matrix of $T$ relative to basis $B=(e_1,e_2,e_3)$ are just the $T(e_1),\ T(e_2),\ T(e_3)$ (column-)vectors, coordinated in $B$.

For example, if $(e_1,e_2,e_3)=(1,x+1,(x+1)^2)$, then we have, by linearity of $T$, $$T(e_2)=T(x+1)=T(x)+T(1)=x+1+1=x+2={\bf 1}\cdot (x+1)+{\bf 1}\cdot 1 \\ T(e_3)=T(x^2+2x+1)=(x+1)^2+2(x+1)+1$$ and you need the coordinates (=coefficients in the linear combination) w.r.t. this same $(e_1,e_2,e_3)$ basis.

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So for example, T(1,0,0)=(1,1,1) ? – Carpediem Feb 18 '13 at 18:40
I don't understand so much your hint – Carpediem Feb 18 '13 at 18:43
No. In your case first $e_1=1,\ e_2=x,\ e_3=x^2$. It is said that $T(1)=1,\ T(x)=x+1,\ T(x^2)=(x+1)^2={\bf 1}\cdot 1+{\bf 2}\cdot x+{\bf 1}\cdot x^2$, so the third column will be $\pmatrix{1\\2\\1}$. – Berci Feb 18 '13 at 18:53
So this is relative to the first basis (1,x,x^2) right ? – Carpediem Feb 18 '13 at 18:56
Can you just confirm my answer? But I still don't know what is the relationship between the two matrices. – Carpediem Feb 18 '13 at 19:03

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