Does this sequences generate that subspace?

Question

Prove that the subspace $W$ of the space of real sequences, formed by the sequences that satisfy $X_{n+2}=X_{n+1} + X_n$, is generated by the sequences $$ v=(1,\alpha,\alpha^2,\alpha^3,\dotsc) \qquad\text{and}\qquad w=(1,\beta,\beta^2,\beta^3,\dotsc). $$

I tried to put a sequence of $W$ as linear combination of $v$ and $w$ ($(X_n)=av+bw$) and solve the linear system to find explicitly $a$, $b$, $\alpha$ and $\beta$ but I couldn't do it.

Answer 1

Govn such a sequence $x$, show that you can find constants $a,b$ such that the first two terms of $x-av-bw$ are zero, i.e., solve $x_0=av_0+bw_0$ and $x_1=av_1+bw_1$ simultaneously. Show that then $x=av+bw$, i.e., $x_n=av_n+bw_n$ for all $n$.