This is linear Algebra Please i need urgent help with this q

This is linear Algebra. Please i need urgent help with this question. i do appreciate clear workout. Thanks.

Find the coordinate matrix of v = 2ax^3 + 3bx^2 + 4c in P_3 with respect to the standard basis B = {x^3, x^2, x, 1}. Find the coordinate matrix of (8,11,10) in P_3 with respect to the standard basis B = {(2,3,4),(4,5,2),(5,2,4)}.

Solution

a) V = 2ax^3 + 3bx^2 + 4c

Standard basis = { x^3 , x^2 , x , 1 }

comparing with standard basis , coordinates of matrix:(2a , 3b , 0 , 4c )

b) ( 8, 11 , 10) = a (2 , 3 , 4) +b( 4, 5 , 2) +c( 5 , 2 , 4)

2a +4b +5c =8 -----(1)

3a +5b +2c = 11 ---- (2)

4a + 2b +4c = 10 -----(3)

Solving the above equation to get a, b, c: we get

a =2 ; b = 1 ; c = 0

coordinate matrix: (a , b, c) ----> (2 , 1 ,0 )