Prove that 1 1 t 1 t 2 and 1 t 3 is a basis for p3

Prove that 1 , ( 1 - t ) , ( 1 - t )² and ( 1 - t )3 is a basis for p⁽³⁾ .

Solution

If we can show that they span standard basis vectors of P3 ie 1,t,t^2,t^3 then the proof is done

Label them as

z0=1

z1=1-t

z2=(1-t)^2

z3=(1-t)^3

z1=1-t

z1=z0-t

t=z0-z1

z2=(1-t)^2=1-2t+t^2=-1+2(1-t)+t^2

z2=-z0+2z1+t^2

t^2=z2-2z1+z0

z3=(1-t)^3=1-t^3-3t+3t^2=z0-3(z0-z1)+3(z2-2z1+z0)-t^3

t^3=z0-3z1+3z2-z3

Hence proved