Euclidean space

Euclidean spaces are generalizations of two- and three-dimensional spaces of Euclidean geometry to arbitrary finite-dimensional spaces. It is the abstraction of a geometric object into a topological and algebraic structure.

Many concepts in algebra and analysis are developed by analogy with geometry, where Euclidean space is among the first few.

## Definitions

Euclidean space is the space of all $n$-tuples of real numbers $(x_1, x_2, \cdots, x_n)$, with inner product $\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_i y_i$. It is commonly denoted as $\mathbb{R}^n$.

Length

Angle

Orthogonal

## Basis

Ortho-normal basis

Gram-Schmidt Orthonormalization, QR decomposition

Orthogonal transformation: rotation, reflection; Improper rotation

## Direct Sum and Space Decomposition

Orthogonal subspace

direct sum

orthogonal complement

## Isometry

Thm: Any finite-dimensional linear inner-product space over the real numbers is isometric with the Euclidean space of the same dimension.