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.


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 $\mathbb{R}^n$.





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


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

🏷 Category=Algebra Category=Linear Algebra