Let u u1 un and v v1 vn be vectors in Rn Define the Eucl

Let u = [u_1 ... u_n] and v = [v_1 ... v_n] be vectors in R^n. Define the Euclidean inner product (I.e. dot product) of u and v, denoted (u, v). Define the Euclidean norm (i.e. 2 norm) of u, denoted ||u||. Define the Euclidean distance between two vectors u and vm denoted dist (u, v).

Solution

The Euclidean inner product or the dot product < u , v> = ( u . v) = u₁ v₁ + u₂ v₂ + …+u_n v_n

b.     A Euclidean norm is a function that assigns a strictly positive length or size to each vector in a vector space except for the zero vector, which is assigned a length of zero. Given a vector space V over a subfield F of the real or complex numbers, a norm on V is a function f : V R with the following properties:

             For all a F and all u, v V,

1.    f(av) = |a| f(v), (absolute homogeneity or absolute scalability).

2.    f(u + v) f(u) + f(v) (triangle inequality or sub- additivity).

3.    If f(v) = 0 then v is the zero vector

            Also, we have f(0) = 0 and f(v) = f(v), so that by the triangle inequality, f(v) 0 (non-negativity).

        The norm of u denoted II u II = ( u₁² + u₂³ +…. + u_n² )

c. The distance from u to v, or from v to u is given by the Pythagorean formula dist( u, v) = [ ( v₁ – u₁)² + (v₂ – u₂ )² +…+ (v_n – u_n)² ]