A 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: Definition
Any Euclidean set, \(\mathbb{R}^n\), together with the canonical vectors space structure: each vector is a column vector, \(\begin{pmatrix} x^1 \\ x^2 \\ ... \\ x^n \end{pmatrix} \in \mathbb{R}^n\); for any \(\begin{pmatrix} x^1 \\ x^2 \\ ... \\ x^n \end{pmatrix}, \begin{pmatrix} x'^1 \\ x'^2 \\ ... \\ x'^n \end{pmatrix} \in \mathbb{R}^n\) and \(r, r' \in \mathbb{R}\), \(r \begin{pmatrix} x^1 \\ x^2 \\ ... \\ x^n \end{pmatrix} + r' \begin{pmatrix} x'^1 \\ x'^2 \\ ... \\ x'^n \end{pmatrix} = \begin{pmatrix} r x^1 + r' x'^1 \\ r x^2 + r' x'^2 \\ ... \\ r x^n + r' x'^n \end{pmatrix}\)
2: Note
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.
