Introduction of basic geometry of Euclidean spaces:

In Euclidean structure, learn euclidean geometry definition helps to take a position O as origin in real n-dimensional Euclidean space. In Euclidean space, the length of vector x represented by || x ||.  The length of vector is the distance to the origin.

 

Let us introduce a Cartesian coordinate system and denote the Cartesian coordinates of x as x1...xn.

In learn euclidean geometry definition, we can use the Pythagorean Theorem frequently we can express the length of x in terms of its Cartesian coordinates.

            || x ||  =  x12+....+xn2                ( 1 )

 

Geometric meaning of (x, y):

 

  In learn euclidean geometry gefinition assisits the scalar product of two vectors x and y, denoted as (x, y), is defined by

                   (x, y) = ∑ xj yj                                                               ( 2 )

We can express the length of a vector as

                  || x || 2 = (x, x).                                                            ( 3 )

The scalar product is commutative:

               (x, y) = (y, x)                                                                    ( 4 )

  And bilinear

                             (x + u, y) = (x, y) + (u, y).

                            (x, y + v) = (x, y) + (x, v).                                      ( 5 )

 Using the algebraic properties of scalar product we can derive the identity

                     (x – y, x – y) = (x, x) – 2(x, y) + (y, y).

 Using (3), we can rewrite this identity as

                      || x – y ||2 = || x ||2 – 2(x, y) + ||y ||2.               ( 6)

The term on the left is the distance of x from y, squared; the first and third terms on the right are the distance of x and y from O, squared.

 

Learn euclidean geometry definition:

 

 The learn euclidean geometry definition, such as

Definition:

A Euclidean geometry in a linear plane X on the real’s is furnished by a real-valued function of two vector arguments labeled a scalar product and denotes as (x, y), which the learn euclidean geometry definition has the following properties:

(a)     (x, y) is a bi linear function; that is, it is a linear function of each argument when the other is kept fixed.

(b)    It is symmetric:

                                                     (x, y) = (y, x).

(c)     It is positive:

                                              (x, x) > 0 except for x = 0.

Defimlion:

The distance of two vectors x and y in a linear space with Euclidean norm is defined as || x - y ||.

Definition:

Let X be a finite-dimensional linear space with Euclidean structure, Y a subspace of  X. The orthogonal complement of Y,  denoted as Y `_|_` , consists of all vectors z in X that are orthogonal to every y in Y :

 z in  Y `_|_` if (y, z)  = 0       for all y  in Y.

 

Euclidean Geometry:

 In learn euclidean geometry definition , we define the Euclidean length (also called norm) of x by   

                                                | x || = (x, x)1/2.A scalar product is also called an inner product, or a dot product.