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? $\boxed{\text{Yes}}$