Exercise I.A.5

Question

Show that the set $\{1,(t-1),(t-1)^2,(t-1)^3\}$ generates $P_3$ the vector space of all polynomials of degree $\leq 3$.

Toğrul Topal · Accepted Answer

Let $p\in P_3$ be arbitrary, we can represent it as $p(t) = a_0 + a_1t_1 + a_2t_2 + a_3t_3$. We want to find coefficients $c_0,c_1,c_2,c_3$ such that
    $$c_0 + c_1(t-1)^2 + c_2(t-1)^2 + c_3(t-1)^3 = a_0 + a_1t_1 + a_2t_2 + a_3t_3$$
    We have
    \begin{align*}
      c_0 + c_1(t-1) + c_2(t-1)^2 + c_3(t-1)^3 &= [c_0] + [c_1t -c_1] + [c_2t^2 - 2c_2t + c_2] + [c_3t^3-3c_3t^2 + 3c_3t - c_3]\
                                               &= c_3 t^3 + c_2 t^2 - 3 c_3 t^2 + c_1 t - 2 c_2 t + 3 c_3 t + c_0 - c_1 + c_2 - c_3\
                                               &= c_3t^3 + (c_2-3c_3)t^2 + (c_1-2c_2+3c_3)t + (c_0 - c_1 + c_2 - c_3)
    \end{align*}
    We thus set
    \begin{align*}
      a_3 &= c_3\
      a_2 &= c_2 - 3c_3\
      a_1 &=c_1 - 2c_2 + 3c_3 \
      a_0 &= c_0 - c_1 + c_2 - c_3 
    \end{align*}
    from which we get
    \begin{align*}
      c_3 &:= a_3\
      c_2 &:= a_2 + 3a_3\
      c_1 &:= a_1 + 2a_2 + 3a_3\
      c_0 &:= a_0 + a_1 + a_2 + a_3
    \end{align*}
    In other words, any polynomial $a_0 + a_1t_1 + a_2t_2 + a_3t_3$ can be represented as a linear combination of the elements of $W$:
    $$(a_0 + a_1 + a_2 + a_3) + (a_1+2a_2+3a_3)(t-1) + (a_2 + 3a_3)(t-1)^2 + a_3(t-1)^3$$

Exercise I.A.5

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Exercise I.A.5

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