definition of Euclidean vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
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:
\( d\): \(\in \mathbb{N} \setminus \{0\}\)
\(*\mathbb{R}^d\): \(= \text{ the Euclidean set }\) with the \(\mathbb{R}\)-scalar multiplication and the addition specified below, \(\in \{\text{ the } \mathbb{R} \text{ vectors spaces }\}\)
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Conditions:
\(\forall r = (r^1, ..., r^d) \in \{\mathbb{R}^d\}, \forall s \in \mathbb{R} (s r = (s r^1, ..., s r^d))\)
\(\land\)
\(\forall r = (r^1, ..., r^d), r' = (r'^1, ..., r'^d) \in \{\mathbb{R}^d\} (r + r' = (r^1 + r'^1, ..., r^d + r'^d))\)
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2: Note
It will be almost obvious that \(\mathbb{R}^d\) satisfies the conditions to be an \(\mathbb{R}\) vectors space: it is closed under the scalar multiplication and the addition; \(0 = (0, ..., 0) \in \mathbb{R}^d\); the additive inverse of \((r^1, ..., r^d)\) is \((- r^1, ..., - r^d)\); \(1 (r^1, ..., r^d) = (r^1, ..., r^d)\) (in fact, this is the archetype of 'vectors space'), so, we omit the rigorous check.
Although the \(\mathbb{R}^n\) 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 \(\mathbb{R}^n\) may have some different structures.
