Question

I need help with number 2 and 3 thank you
Please show all steps
much appreciated

2. Write the co ordinate v in R (with usual basis) in terms of the given basis S answer. 3. Determine whether the set S-$3,x, x-x spans P. Explain your

Answer 1

v = (2, -1, 4) s = {(1, 0, 1), (1, 1, 0), (0, 1,1)} Let (2, -4, 4) = a(1, 0, 1) + b(1, 1, 0) + c(0, 1, 1) Rightarrow a + b = 2 Rightarrow a - c = 3 b + c = -2 a + c = 4 Rightarrow 2a = 7 Rightarrow a = 7/2, c = 1/2 b = -5/2 Rightarrow v = 7/2 (1, 0, 1) - 5/2 (1, 1, 0) = 1/2 (0, 1, 1) So The coordinate are (7/2, - 5/2, 1/2) Now lt is a finite basis we will check is it linearly independent or not, so it 3a + bx + c (x -X^2) = 0 Rightarrow 3a + (b + c) x -cx^2 = 0 Rightarrow 3a = 0 Rightarrow a = 0, c = 0 b + c = 0 Rightarrow b = 0 Rightarrow a = b = c = 0 Rightarrow S is