Difference between revisions of "Euclidean space"
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| − | The '''Euclidean space''' of dimension <math>n</math> refers to the set of points <math>(x_1, x_2, \ldots, x_n)</math>, where each <math>x_i</math> is a real number. The two-dimensional Euclidean space is the [[Cartesian plane]], and so forth. | + | The '''Euclidean space''' of dimension <math>n</math> refers to the [[set]] of points <math>(x_1, x_2, \ldots, x_n)</math>, where each <math>x_i</math> is a [[real number]]. The two-dimensional Euclidean space is the [[Cartesian plane]], and so forth. In this representation, the study of Euclidean space is called [[coordinate geometry]]. |
| − | + | Euclidean space can also be viewed as an example of one of several more general mathematical objects. For example, Euclidean space is a [[metric space]] with respect to the [[usual distance formula|distance]] [[metric]], <math>d(\bold{x},\bold{y}) = \sqrt{(x_1-y_1)^2 + \cdots + (x_n - y_n)^2}</math>. Similarly, the Euclidean space of dimension <math>n</math> is the unique (up to [[isomorphism]]) [[vector space]] of dimension <math>n</math> over <math>\mathbb{R}</math> (under [[pointwise addition]] and the "usual" scalar multiplication <math>c \cdot (x_1, \ldots, x_n) = (c x_1, \ldots, c x_n)</math> for <math>c, x_i \in \mathbb{R}</math>). | |
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[[Category:Geometry]] | [[Category:Geometry]] | ||
Revision as of 11:46, 13 March 2010
The Euclidean space of dimension 
refers to the set of points 
, where each 
is a real number. The two-dimensional Euclidean space is the Cartesian plane, and so forth. In this representation, the study of Euclidean space is called coordinate geometry.
Euclidean space can also be viewed as an example of one of several more general mathematical objects. For example, Euclidean space is a metric space with respect to the distance metric, 
. Similarly, the Euclidean space of dimension is the unique (up to isomorphism) vector space of dimension over 
(under pointwise addition and the "usual" scalar multiplication 
for 
).
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