Question

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)

Answer 1

6 Ca Zet B = (bi, be . by be basis of over F. Then, o a vector de ev can be expressed as = ki bet kabet - et knibh The Co-ordinate vector Cki, kg.-kn) if it expressed as a Coloumh- vector is denoted by ki ? Kn T2,B] og [2079 PC = 3+x - 2x² + 463 The given basis B are - B - ( 1, x, x 3)

Now, 3+*+2017+4x3 = a (1) +3 CC) to cc td C 3+x - 2167483 a+bx+coc? + da? for two polynomials to be equal the Coefficients of each Power of x must be equal. So, a=3 3- 1 ca-2, d = 4 ; Co-ordinates vector of Pex Writ. the given basis are - 37 POU B 1 -2 4

(c) P(x) = 3+2 – 2x tus oli The given basis or are - 8 = (1, 1-X 1-ste, le tex3] Now, 3+x-2xt uses = aci) + b (1-x) +Ccrets 2) td Clx+ 13) 3+Jc2x²+48c3 = atb-bit c-catch td-dxtdxdx3 3t-axt ux? (atbtctd) + (-b-c-d}x +(etd) x - dx3 Poly Domials to be equal, the Coefficients of each Power of x must be equal for two So, atbtctd = 3

-b-C-d = 1 eld = -2 वडप - (d=-4 c+d=- [c=2 and And -b-c-d=17 16 = 1 aul at&tetel = 3 => a=4 Co-ordinates vector of P(x) wisat the given basis 4 fpex] g = 1 2. -4