How do I generalize a $m * n$ dimensional megapixel image as a set?

Question

I have been trying to wrap my head around visualizing high-dimensional vector spaces as a state space with many degrees of freedom, and while I feel like I understand it abstractly, I cannot really write it in set-builder notation. Like for a megapixel image, assuming every pixel is stored as an 8-bit color, how can a $m * n$ image be generalized in a set-builder notataion?

Answer 1

You can think of your image as a list of $mn$ numbers. This throws away the rectangular arrangement of the pixels, but that is OK. If you want a vector space over a finite field, we can imagine that each pixel could take a value from $0$ through $256$ as $257$ is a prime. This gives us a vector space of dimension $mn$. You can add two vectors and multiply a vector by a scalar from your field, just as required. You will satisfy all the necessary commutative, associative, and distributive properties because you are working in a field.