Understanding Linear Transformations | Change of Basis

So, towards the end of the video, I wasn't able to provide a plausible explanation as to why the new matrix we formed, or why the coordinated of the vectors form the transformation. I figured out why, but I need the viewers to figure it out themselves.
A hint is that just look at how we write the transformations when we use the standard basis. And we just changed the basis, so now we need to look at the coordinates in a similar way.
@4:53 i depicted the point (2,4) incorrectly. Apologies.
If you still have doubts, ask me in the comments.