Let $\mathcal{P}_{2}$ denote the vector space of all polynomials in the variable $x$ of degree less than or equal to 2 . Let $B=\left\{1, x, x^{2}\right\}$ be an ordered basis for $\mathcal{P}_{2}$. Let []$_{B}: \mathcal{P}_{2} \rightarrow \mathbb{R}^{3}$ be the linear transformation determined by
\[
[1]_{B}=\vec{e}_{1}, \quad[x]_{B}=\vec{e}_{2}, \quad\left[x^{2}\right]_{B}=\vec{e}_{3} .
\]

Find the coordinate vector representation for each of the following polynomials. Your answers should be vectors of the general form < 1,2,3> .
a. $[4]_{B}=$
b. $\left[6+8 x^{2}\right]_{B}=$
c. $\left[3 x^{2}+7 x-2\right]_{B}=$
d. Is the linear transformation []$_{B}$ an isomorphism? choose

Step 1 ：The question is asking for the coordinate vector representation of the given polynomials with respect to the basis B. The basis B is given as {1, x, x^2}. This means that any polynomial in P2 can be represented as a linear combination of these basis vectors. The coefficients of this linear combination will give us the coordinate vector representation of the polynomial.

Step 2 ：For example, for the polynomial 4, it can be represented as 4*1 + 0*x + 0*x^2. Therefore, its coordinate vector representation is <4,0,0>.

Step 3 ：We can apply the same logic to find the coordinate vector representation of the other polynomials.

Step 4 ：The last part of the question asks if the linear transformation []_B is an isomorphism. A linear transformation is an isomorphism if it is both injective (one-to-one) and surjective (onto). Since the transformation is defined from P2 to R^3, and both P2 and R^3 have the same dimension (3), the transformation is an isomorphism.

Step 5 ：Let's calculate the coordinate vector representation for each polynomial and confirm that the transformation is an isomorphism.

Step 6 ：Final Answer: a. $[4]_{B}=oxed{<4,0,0>}$ b. $\left[6+8 x^{2}\right]_{B}=oxed{<6,0,8>}$ c. $\left[3 x^{2}+7 x-2\right]_{B}=oxed{<-2,7,3>}$ d. Is the linear transformation []$_{B}$ an isomorphism? $\boxed{\text{Yes}}$