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I am trying to understand a question from a test that uses the term "256-dimensional Euclidean orthonormal hyperspace". I don't know what that term means (my math knowledge is only high school level). The test says one may need "to make small queries about the meaning of some words", but Google did not help. So I ask if the meaning of this term can be explained for someone with just knowledge of high school math (maybe there is some YouTube video explaining?).

Thank you!

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    $\begingroup$ By itself that phrase does not convey a problem statement, and it is possibly not desirable to share what the problem to be solved is. So the best suggestion I can make (having followed your test link above) is to assume the most ordinary rendering of that phrase. You can then check that interpretation against the problem statement to see if it makes sense. $\endgroup$
    – hardmath
    Aug 13, 2022 at 17:39

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  • Hyperspace to me suggests simply a space with more than your regular 3 dimensions. There are more complicated concepts using the same term, some of which might even apply to the space in question, but shouldn't be too relevant.
  • Euclidean space means something with basic geometric properties. In layman's terms, perhaps the most relevant aspects would be the ability to measure lengths and angles. It also implies the user of real numbers for distances or coordinates.
  • Vector space isn't explicitly mentioned in your phrase, but I think it's kind of implied by the other terms, “orthonormal” in particular. So while you might have different approaches for how to describe an Euclidean space, using vectors is common.
  • Orthonormal is not really a property of the space itself, but rather if its basis (or coordinate system). But in layman's terms that distinction might not be relevant. In short this means that if you use coordinates to describe your points, then the coordinate axes are perpendicular (i.e. orthogonal) to one another. Furthermore if you change a single coordinate number by one, the corresponding points are one distance unit apart. The lengths in your coordinate system are normalized.

Taking it all together, the intuition I'd suggest is this:

  • Each point in your space is simply a sequence of 256 real numbers.
  • If you want you can compute the distance between two points using Pythagoras.
  • If you want you can compute an angle between lines, using tools such as inner product and arc cosine.
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