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I was studying some machine learning algorithms and I noticed something confusing; hence, I am asking these questions:

1) What is the difference between Euclidean space and vector space?

2) For example, I have a data sample which consists of 7 different attributes. To work with this sample, I work in R^7 space.

In this case, is R^7 a vector space or Euclidean space? If it is not Euclidean space, which dimensions are covered by Euclidean space ?

Goktug
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  • See the definition of Euclidean Space. It is a real vector space. So a vector space over some other field is not Euclidean. Moreover, if there is no inner product, then the space also is not Euclidean. – Dietrich Burde Apr 01 '18 at 16:19
  • In brief: $\mathbb{R}^n$ is called Cartesian space; $n$ can be any dimension. "Euclidean space" is used when we ignore coordinates and instead focus on geometric shapes --- points, lines, triangles, etc. "Vector spaces" (over the real numbers) are used when we ignore coordinates and instead focus on vector sum and scalar multiplication (with real number scalars). "Euclidean space" is often used as a synonym for Cartesian space and for vector space, although ordinarily neither Euclidean spaces nor vector spaces have a preferred coordinate system. – Lee Mosher Apr 01 '18 at 16:32
  • @LeeMosher at first thank you for your explanation. I heard something like this : "In non-euclidean space, mean point of set cannot be calculated." This sounds to me weird because non-euclidean space is a new to me. – Goktug Apr 01 '18 at 16:37
  • Euclidean spaces are a sub-class of the very broad class of things called vector spaces. – DanielWainfleet Apr 01 '18 at 20:18

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I assume you know the definition of vector space. As you probably know, $\mathbb{R}^n$ is a vector space.

Now, if $K$ is a field, an euclidean space is a $K$-vector space $V$ where you have a notion of "positive definite inner product", that is to say, a bilinear, symmetric form $\phi:V\times V\to K$ such that for each $v\in V$, $\phi(v,v)\geq0$ and $\phi(v,v)=0 \iff v=0$.

You can check by exercise that the function $\cdot:\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}$ defined by $\mathbf{x}\cdot\mathbf{y}=\sum_{i=1}^{n}x_iy_i$ is a bilinear symmetric form, and it has the "positive definite" property mentioned above. This makes $\mathbb{R}^n$ an euclidean vector space.

We can define the "norm" $\Vert \mathbf{x}\Vert$(i.e. the distance from the "origin") of a vector to be the product $\mathbf{x}\cdot\mathbf{x}$, and pretty much recreate euclidean geometric concepts such as angles, parallelism and orthogonality by defining the distance between two points to be $\Vert\mathbf{x}-\mathbf{y}\Vert$.

If you want, you can think of an euclidean space as a space made of points and lines etc., while a vector space does not necessarily resemble such a specific structure.

TooAbstract
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