Can someone explain to me why when two transformations match at infinitely many points, something non-singular is happening?

And why if two transformations match at infinitely many points, the difference is 0 at many points?

What is the “transformation” that the lecturer meant at this point? Is that the matrices or the transformation from a basis to another basis ?

I also don’t understand this. I almost think the instructor meant “something singular” instead of “something non-singular”.

I’m incredibly confused about this part. In the next slide, the instructor calls the 2 and 3 as eigenvalues right at the beginning, but I don’t understand how he would know that at that point (before getting to the end of the slide, where he shows the characteristic polynomial).

Also, the matrix

```
2 1
0 3
```

keeps being used as an example, then we get to the conclusion that the eigenvalues are 2 and 3. Is that because 2 and 3 are in the diagonal of the matrix, or is that just a coincidence? I’m understanding this because the diagonal entries represent the stretch in each axis, so it makes sense. But if it is already in the matrix, I don’t understand why we need to “find” them.

The same thing with the result of the characteristic polynomial, which is `(lambda - 2) * (lambda - 3)`

, and the results are 2 and 3, is that always the case, or just a coincidence? For instance, if it was `(lambda - a) * (lambda - b)`

, would the results be `a`

and `b`

?

If these 2 are just coincidences, I would say that’s a really unfortunate example to be used in the course.