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**Isomorphism** Dont worry about it just because its a polynomial. After all $\displaystyle a_0 + a_1x + a_2x^2$ can be viewed as the tuple $\displaystyle (a_0, a_1, a_2)$ with B as basis. With this look it is just another $\displaystyle \mathbb{R}^3$ vector, isnt it?

So the transformation looks like $\displaystyle T[p(x)] = p(2x+1)$ and we want to find the matrix representation for this, right?

$\displaystyle T (a_0 + a_1x + a_2x^2) = a_0 + a_1(2x+1) + a_2(2x+1)^2$

$\displaystyle a_0 + a_1(2x+1) + a_2(2x+1)^2 = a_0 + 2a_1x+a_1 + 4a_2 x^2 + 4 a_2 x +a_2$

$\displaystyle a_0 + 2a_1x+a_1 + 4a_2 x^2 + 4 a_2 x +a_2 = (a_0 + a_1 + 4a_2) +(2a_1 + 4a_2)x + (4a_2)x^2 $

So $\displaystyle a_0 + a_1x + a_2x^2$ gets mapped to $\displaystyle (a_0 + a_1 + 4a_2) +(2a_1 + 4a_2)x + (4a_2)x^2$.

In terms of triples, its the same as saying $\displaystyle (a_0, a_1, a_2)$ maps to

$\displaystyle (a_0 + a_1 + 4a_2, 2a_1 + 4a_2 , 4a_2)$

So now this problem is the same as the first question :)

Just a\find a 3 x 3 matrix that maps $\displaystyle (a_0, a_1, a_2)$ maps to

$\displaystyle (a_0 + a_1 + 4a_2, 2a_1 + 4a_2 , 4a_2)$.