Opening the Black Box: Recurrence Relations and linear algebra.

This video series provides a good foundation for an intuition of Linear algebra: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
And this specific video provides the tools to solve recurrence relations using linear algebra: https://youtu.be/PFDu9oVAE-g

Spoilers to the end of the video:

It presents a problem of finding the closed form of the fibannoci series in terms of:

let $ f_n $  be the nth fibanocci number

\[ A = \begin{pmatrix} 1 & 1\\ 1 & 0 \end{pmatrix}\quad and \quad \vec{F_n} =\begin{pmatrix} f_{n} \\ f_{n-1} \end{pmatrix} \]

Then the recurrence rule can be written as: \[ \vec{F_{n}} = A\vec{F_{n-1}} \]

And the closed form: \[ \vec{F_{n}} = A^n\vec{F_{1}} \]

And eigenvectors can be used to efficiently calculate using the closed form.