I'm having difficulty with the following, problem 5.3.7 from Gilbert Strang's "Linear Algebra and its Application".
The numbers $\lambda_1^k$ and $\lambda_2^k$ satisfy the Fibonacci rule $F_{k+2}=F_{k+1} + F_k$ :
$\lambda_1^{k+2}=\lambda_1^{k+1} + \lambda_1^k$ and $\lambda_2^{k+2}=\lambda_2^{k+1} + \lambda_2^k$
Prove this by using the original equation for the $\lambda's$ (Multiply it by $\lambda^k$)
Then any combination of $\lambda_1^k$ and $\lambda_2^k$ satisfies the rule.
The combination $F_k=(\lambda_1^k-\lambda_2^k)/(\lambda_1-\lambda_2)$ gives the right start of $F_0=0$ and $F_1=1$.
I'm not sure what "the original equation for the $\lambda's$" is, or what I'm supposed to prove. Can someone please help?