Difference equations $u_{k+1} = Au_{k}$ with Linear algebra

I'm newbie in Linear Algebra and have no knowledge in Differential equation. I'm studying Introduction to Linear algebra https://www.youtube.com/watch?v=13r9QY6cmjc in the video (at 6:00) the professor says $$AS = S \Lambda$$

Where A is a square matrix that has n independent eigenvector and S is a matrix of independent eigenvectors of A

then later in the video (at 27:30) when he solve $$u_{k+1} = Au_{k}$$

he goes into $$u_{k} = A^ku_{0}$$

Where $$u_{0} = c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}+...c_{n}x_{n} = Sc$$

Then $$u_{k} = A^kSc = S\Lambda^kc$$ isn't it? why he writes $A^{100}u_{0} = \Lambda^{100}Sc$ it suppose to be $A^{100}u_{0} = S\Lambda^{100}c$ isn't it?

Is he write the wrong equation? and the $A^{100}u_{0} = S\Lambda^{100}c$ is the correct one?

I see a lot of youtube comment said he was wrong on this but his document still make the same mistake as you can see on this link on page 2 in the section "Difference equations $u_{k+1} = Au_{k}$" the last line of the section writes $u_{k} = A^ku_{0} = c_{1}λ_{1}^kx_{1} +c_{2}λ_{2}^kx_{2} +···+c_{n}λ_{n}^kx_{n} = Λ^kSc$

which I think is wrong. It should be:

$u_{k} = A^ku_{0} = c_{1}λ_{1}^kx_{1} +c_{2}λ_{2}^kx_{2} +···+c_{n}λ_{n}^kx_{n} = SΛ^kc$

Am I right?

You are quite right (and the video is wrong on this). The point is that acting with $A$ upon the eigenvectors produce the eigenvectors times eigenvalue. And this written in matricial form is precisely $$A S = S \Lambda$$ as you say. Similarly, $A^{100} S = S \Lambda^{100}$ etc...