The question: [Matrix dot product][1] [Projection in dot product][2] Why are these two equal?
My work till now. It may help: I have attached images because pdf doesn't work, but still I can not poste more than 10 of them, so here is a google drive link to a pdf I made: https://drive.google.com/file/d/1XWleXEybn1JoPCUzHG49RoK060CCfUNS/view?usp=sharing
I found the answer myself. The thing a was looking for is called duality. But more precisely what I was missing is what you need to take a number line where the $\vec{v_1}$ lies. And then define $\hat{u}$ which is kind of the unit vector in that number line. So logicaly $\vec{v_1}$ = [some number] x $\hat{u}$.
Now the idea is to squish the 2D space into this number line with a linear transformations. You find where the $\hat{i}$ and $\hat{j}$ land, which I called $\hat{\hat{i}}$ and $\hat{\hat{j}}$ respectively. And construct a linear transformation matrice where $\hat{\hat{i}}$ and $\hat{\hat{j}}$ are represented with the $\hat{u}$ unit vector. Because they lie in the same line as $\hat{u}$ there is some nice symmetry between the $\vec{v_1}$ components $\vec{v_x}$,$\vec{v_y}$ and $\hat{\hat{i}}$,$\hat{\hat{j}}$. Unfortunately I cannot really draw here. But the 3blu1brown video on dot product and duality does that amazingly clear. Because of this symmetry, the linear transformation matrix looks like this [$v_x$ $v_y$]. Now you multiply this with the $\vec{v_2}$ and you have the tools to describe projection with linear transformations.
The projection happens because when you squish 2D in the number line with this matric [$v_x$ $v_y$], the $\hat{i}$ and $\hat{j}$ get projected into the number line and form $\hat{\hat{i}}$ and $\hat{\hat{j}}$, which lie on the number line. Because $\hat{i}$ and $\hat{j}$ get projected all the other vectors represented by them, like $\vec{v_2}$, do too.
Link to the 3blu1brown video that helped me: https://youtu.be/LyGKycYT2v0
Notes. What I did in my pdf is take this number line in the x axes (where $\hat{i}$ lies), which is not a good idea, because you don't see the projection.