The texture map used here is a map that (0,0) is black and (1,1) is yellow. We apply the texture map to everywhere or in other words the texture coordinate repeats in the entire image.

How do we determine the coordinates of vertices of squares in the sample space?

For example, I imagine that we can have much bigger squares in the sample space than those in the current image. Where does the rendering system store this information that determines the size and location of squares?

I feel like the size and location of these “squares” are not stored explicitly per se, but rather implicitly. If you look at slide 9 and 10, they say that the texture coordinates are stored for the vertices of the triangles (and that the texture coordinates of the points inside a triangle needs to be calculated through interpolation). So the size and locations of these texture squares are actually stored implicitly by the texture coordinate info of the triangle vertices.