The span describes the entire space. It's a set of vectors that you can combine using addition and multiplication in order to obtain any other vector in the space. For example a spanning set over the real number plane would be {(1,0), (0,1)}. This particular set is also an orthonormal basis and you can think of each vector as representing two orthogonal dimensions. This is because their dot product is 0.

However, any set of two vectors that are not on the same line will span the real number plane. For example, {(1,1), (0,1)} spans the real number plane, but they are not orthogonal.

Overall though it is always important to be aware of your input space, and the features/dimensions that you use to represent it. You can easily introduce bias or just noise in a number of ways if you aren't thorough. One example would be not normalizing your data.

>so the extra dimensions are unnecessary

Yes one reason for embedding is to get extract relevant features.

Also, any finite dimensional inner product space has an orthonormal basis, and the math is easiest this way so there's not much of a reason to describe a space using non orthogonal dimensions. There is also nothing stopping you from doing so though.

>Doesn't it suggest a pattern in data if a mapping is found that reduces dimension

Yeah generally you wouldn't attempt to use ML methods on data where you think there is no pattern

>Something something Linear algebra

I think you might be thinking about the span and or basis but it's hard for me to interpret your question

Embedding is a way to map the high dimension vectors in your input space to a lower dimension space.

Pick any tautology and write it down in any language you choose. That statement will be always be true regardless of what symbols you use to represent it.

Logic is universal it has nothing to do with being a human.

It's not a valid comparison because mathematics is based on statements that are self evidently true. The language we use to describe those truths is completely independent from the meaning behind the symbols.

A sufficiently advanced species is more than likely going to have a syntax that is completely alien to us. However, they would certainly be describing many of the same things that we do.

Think about something fundamental like a circle. I don't care what language or symbols you use to describe it, because if you gave me enough time I would be able to tell exactly what you mean. The concept and meaning behind a circle exists independently of the syntax used to describe it.

The syntax of mathematics is independent from the semantics.

Probably referencing the radio signals it emits