EuclideanSpace

Euclidean Space

1. Definition:
   A Euclidean space ( \mathbb{R}^n ) is a set of all ordered n-tuples of real numbers. An element ( x ) in ( \mathbb{R}^n ) is represented as: [ x = (x_1, x_2, \dots, x_n) ] where each ( x_i ) is a real number.

2. Distance:
   The distance between two points ( x = (x_1, x_2, \dots, x_n) ) and ( y = (y_1, y_2, \dots, y_n) ) in ( \mathbb{R}^n ) is given by the Euclidean distance formula: [ d(x,y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \dots + (x_n - y_n)^2} ]

3. Inner Product:
   The dot product (or scalar product) of two vectors ( x ) and ( y ) in ( \mathbb{R}^n ) is defined as: [ x \cdot y = x_1y_1 + x_2y_2 + \dots + x_ny_n ]

4. Norm:
   The norm (or length) of a vector ( x ) in ( \mathbb{R}^n ) is defined in terms of the dot product: [ ||x|| = \sqrt{x \cdot x} = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2} ] This is essentially the distance from the origin to the point represented by the vector ( x ).

5. Geometry:
   In a Euclidean space, the geometric notions of lines, angles, planes, and other shapes can be defined and studied. For instance, the angle θ between two vectors ( x ) and ( y ) can be found using the dot product: [ \cos(θ) = \frac{x \cdot y}{||x|| \cdot ||y||} ]

6. Linear Transformations:
   Transformations in Euclidean space that preserve vector addition and scalar multiplication are called linear transformations. They can often be represented by matrices. For example, in ( \mathbb{R}^2 ), a transformation ( T ) can be represented by a 2x2 matrix ( A ) such that: [ T(\mathbf{v}) = A\mathbf{v} ] where ( \mathbf{v} ) is a vector in ( \mathbb{R}^2 ).

7. Orthogonality:
   Two vectors are orthogonal (perpendicular) if their dot product is zero. For ( x ) and ( y ) in ( \mathbb{R}^n ): [ x \cdot y = 0 ] implies that ( x ) is orthogonal to ( y ).

In summary, Euclidean space is the familiar setting for multivariable calculus, linear algebra, and other areas of mathematics. It's a framework where the geometric and algebraic notions align well, making it a central object of study in many fields.