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asdf60
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I understand that the definition of the number of dimensions of a vector space, but somehow that doesn't really help me with physical dimensions. How in practice do we know that our space is 3-dimensional?
How in practice do we know that our space is 3-dimensional?
I actually think that's a really good question. It's probably because we generally assume space is homogeneous and isotropic, and since we've always been able to describe the positions of everything with three numbers we assume it works everywhere in the universe. There is no reason to believe there are more than three macroscopic spatial dimensions, so there's no reason to have a physical model that uses any number of macroscopic spatial dimensions than three. The assumption that space is homogeneous and isotropic dates back to Galileo, and so far it's proven to be a valuable postulate.asdf60 said:I was looking for that answer...because I'm a bit skeptical about how. How do we prove that we can unambiguously specify any (and every) point in space with just 3 coordinates?
reilly said:We use three dimensions because it seems to work for everyday life. Check out a topographical map -- they always hit the nail on the head, well almost always. And, we really can't draw a 4 or higher dimensional object. Nature makes our perceptions intelligible in three (or less) dimensions; why? Who knows. (The extra dimensions of string theory are just that, theoretical concepts. )
Regards,
Reilly Atkinson
asdf60 said:I understand that the definition of the number of dimensions of a vector space, but somehow that doesn't really help me with physical dimensions. How in practice do we know that our space is 3-dimensional?
esources: "dimension theory" by hurewicz and wallman, "why space has 3 dimensions" by poincare.
poincare's essay is for the general public on the notion of dimension. he says basically that he calls a finite set zero dimensional for starters. then a set is 1 dimensional if it can be separated by removing a zero dimensional set. e.g. as matt grime pointed out, R^1 is disconnected by the removal of anyone point, hence is one dimensional.
R^2 is not disconnected by removing one point, but is disconnected by removing a copy of R^1 hence R^2 is two dimensional. etc etc..
The number of dimensions of a vector space refers to the minimum number of linearly independent vectors that are needed to span the entire space.
The number of dimensions of a vector space can be determined by finding the maximum number of linearly independent vectors that can be found in the space.
Yes, a vector space can have an infinite number of dimensions. This is often seen in spaces such as the real numbers or complex numbers.
The number of dimensions of a vector space is the minimum number of linearly independent vectors needed to span the entire space, while the number of basis vectors is the actual number of vectors that form a basis for the space.
The number of dimensions affects the operations performed in a vector space by defining the maximum number of dimensions that a vector can have and the maximum number of linearly independent vectors that can be used in operations such as addition and multiplication.