Are the vectors 15 15x 15x2 6 3x 9x2 and 21 18x 6x2 l

Are the vectors 15 + 15x - 15x^2, 6 + 3x + 9x^2 and - 21 - 18x + 6x^2 linearly independent? If the vectors are independent, enter zero in every answer blank since zeros are only the values that make the equation below true. If they are dependent, find numbers not all zero, that make the equation below true. You should be able to explain and justify your answer. 0 = (15 + 15x - 15x^2) + (6 + 3x + 9x^2) + (-21-18x + 6x^2).

Solution

vectors are : [15 15 -15]^T, [6 3 9]^T, [-21 -18 6]^T

suppose vectors are linderly dependent

so some non-zero value should be exist for a,b,c where are are satisfying these equation

15a + 6b -21c =0 -----------------------1

15a +3b -18c =0 -----------------------2

-15a +9b + 6c =0 ----------------------3

eqn 2 - eqn 1 : b =c

eqn 1 - 2eqn 2 : -15a +15c = 0 : b =c =a

so using eqn 1: a = b= c= 0

so our assumption is wrong ; vectors are independent