2023-12-10

426: Euclidean Vectors Space

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definition of Euclidean vectors space

Topics


About: vectors space

The table of contents of this article


Starting Context



Target Context


  • The reader will have a definition of Euclidean vectors space.

Orientation


There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.


Main Body


1: Structured Description


Here is the rules of Structured Description.

Entities:
d: N{0}
Rd: = the Euclidean set  with the R-scalar multiplication and the addition specified below, { the R vectors spaces }
//

Conditions:
r=(r1,...,rd){Rd},sR(sr=(sr1,...,srd))

r=(r1,...,rd),r=(r1,...,rd){Rd}(r+r=(r1+r1,...,rd+rd))
//


2: Note


It will be almost obvious that Rd satisfies the conditions to be an R vectors space: it is closed under the scalar multiplication and the addition; 0=(0,...,0)Rd; the additive inverse of (r1,...,rd) is (r1,...,rd); 1(r1,...,rd)=(r1,...,rd) (in fact, this is the archetype of 'vectors space'), so, we omit the rigorous check.

Although the Rn set tends to be implicitly supposed to have some canonical structures (including the Euclidean vectors space structure), it is not necessarily so. It is important to be aware that Rn may have some different structures.


References


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