426: Euclidean Vectors Space

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A definition of Euclidean vectors space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of %field name% vectors space.

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Target Context

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

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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.

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Main Body

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1: Definition

Any Euclidean set, ${\mathbb{R}}^n$, together with the canonical vectors space structure: each vector is a column vector, $\left(\begin{array}{c}{x}^1\\ x^2\\ ...\\ x^n\end{array}\right)\in {\mathbb{R}}^n$; for any $\left(\begin{array}{c}{x}^1\\ x^2\\ ...\\ x^n\end{array}\right),\left(\begin{array}{c}{x}^{\prime 1}\\ x^{\prime 2}\\ ...\\ x^{\prime n}\end{array}\right)\in {\mathbb{R}}^n$ and $r,r^{\prime }\in \mathbb{R}$, $r\left(\begin{array}{c}{x}^1\\ x^2\\ ...\\ x^n\end{array}\right)+r^{\prime }\left(\begin{array}{c}{x}^{\prime 1}\\ x^{\prime 2}\\ ...\\ x^{\prime n}\end{array}\right)=\left(\begin{array}{c}r{x}^1+r^{\prime }{x}^{\prime 1}\\ r{x}^2+r^{\prime }{x}^{\prime 2}\\ ...\\ r{x}^n+r^{\prime }{x}^{\prime n}\end{array}\right)$

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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.

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References

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