# 6. (a) Let V be a vector space over the scalars F, and let B =…

6. (a) Let V be a vector space over the scalars F, and let B = (01.62, …, On) CV be a basis of V. For v € V, state the definition of the coordinate vector [v]s of v with respect to the basis B. [2 marks] (b) Let V = R$[x] = {ao + a11 + a222 + a3r | 20, 41, 42, 43 € R} the vector space of real polynomials of degree at most three. Write down the coordinate vector of the polynomial p(x) = 3 + 1 – 2x² + 4x) with respect to the basis B = (1,4,22,23). [2 marks] (c) Now again let V = R3[]. Write down the coordinate vector of the same polynomial p(x) = 3 + 1 – 2x² + 4.23 but this time with respect to the basis B = (1,1-2, 1-1+1?, 1 – 2 + ? – 29). [6 marks)