(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the functions p_{o}(t)=1, p_{1}(t)=t, p_{2}(t)=1/2(3t^{2}-1), and p_{3}(t)=(3/2)*[(5/3)t^{3}-t) also form a basis for the vector space P^{3}(R) ... "R" meaning all real numbers

2. Relevant equations

I know these polynomials are the first four Legendre polynomials

3. The attempt at a solution

I know that proving functions form a basis involves proving that each funciton has a unique representation as a linear combination... I'm not certain on what this means exactly but I'm trying my best to figure it out... heres my work so far on the problem

a(1) + b(t) + c/2(3t^{2}-1) + d/2(5t^{3}-3t) = a + bt + ct^{2}+ dt^{3}

at t=0 ; -c/2 = 0 ; so c=0

a + b(t) + d/2(5t^{3}-3t)=0

at t=0 ; a=0

at t=1 ; b=-d

at t=2 ; -2d + d/2(34) = 15d = 0 ; d=0 ; b=0

so the legendre polynomials are a linearly independent set at; 1/2 + t - 3/2t + 3/2t^{2}+5/3t^{3}= 1/2-1/2t+3/2t^{2}+5/3t^{3}

; a polynomial in the span of P(R) that forms a basis

thanks in advance for any advice you have to offer me

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# Homework Help: Showing functions form a basis

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