Does #S={t^3-t^2+1,t^2-4, t^3+2t , 5t}# generate #P3#? ( #P3# means polynomials with degree #3#)
1 Answer
Feb 25, 2018
Yes
Explanation:
Writing the coefficients of these polynomials as a matrix and taking the determinant, we find:
#abs((1, -1, 0, 1),(0, 1, 0, -4),(1, 0, 2, 0),(0, 0, 5, 0))#
#= abs((1, -1, 0, 1),(0, 1, 0, -4),(0,1,2,-1),(0,0,5,0))#
#= abs((1, -1, 0, 1),(0, 1, 0, -4),(0,0,2,3),(0,0,5,0))#
#= -abs((1, -1, 0, 1),(0, 1, 0, -4),(0,0,5,0),(0,0,2,3))#
#= -abs((1, -1, 0, 1),(0, 1, 0, -4),(0,0,5,0),(0,0,0,3))#
#= -15#
Since the determinant is non-zero, these polynomials are linearly independant and do generate
