Matrices and basis 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

**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.