ABSTRACT
We know that the operations addition and scalar multiplication in Euclidean space R n $ {\mathbb{R}}^{n} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315172309/4e6ab8d8-ae27-4ebf-9151-a8c1ccdef5b3/content/inline-math3_1.tif"/> produce vectors within R n $ {\mathbb{R}}^{n} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315172309/4e6ab8d8-ae27-4ebf-9151-a8c1ccdef5b3/content/inline-math3_2.tif"/> . Namely R n $ {\mathbb{R}}^{n} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315172309/4e6ab8d8-ae27-4ebf-9151-a8c1ccdef5b3/content/inline-math3_3.tif"/> is closed under addition and scalar multiplication. In addition, the derived operations, such as exchange of order of addition, do not produce a vector different from the one before the operation. To be specific, for u, v, w ∈ R n , $ w \in {\mathbb{R}}^{n} , $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315172309/4e6ab8d8-ae27-4ebf-9151-a8c1ccdef5b3/content/inline-math3_4.tif"/> s, t ∈ R , $ t \in {\mathbb{R}}, $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315172309/4e6ab8d8-ae27-4ebf-9151-a8c1ccdef5b3/content/inline-math3_5.tif"/> R n $ {\mathbb{R}}^{n} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315172309/4e6ab8d8-ae27-4ebf-9151-a8c1ccdef5b3/content/inline-math3_6.tif"/> satisfies the following properties:
