Properties of Algebraic Operations

By definition, a collecton of vector objects is called a real vector space only if the defined addition and scalar multiplication operations for all given vector objects satisfy all typical properties of addition and scalar multiplication operations for a real vector space. These typical properties are fundamental laws of 𝑛-tuples for addition and scalar multiplication operations used in real vector space.

Algebraic Laws for Scalar Multiplication

Let set 𝑆 be an 𝑛-Tuple Vector Space and 𝛼, 𝛽 are scalars in ℝ. The 𝑛-tuples of set 𝑆 also satisfy some fundamental algebraic laws for the multiplication operation by a real number scalar. That is

Closure Law of Scalar Multiplication: If 𝑨∊𝑆, 𝛼∊ℝ, then 𝛼𝑨∊𝑆
Scalar Identity of Scalar Multiplication: Scalar Identity 1∊ℝ:1𝑨=𝑨
Vector Distribution Law of Scalar Multiplication: (𝛼+𝛽)𝑨=𝛼𝑨+𝛽𝑨
Scalar Distribution Law of Scalar Multiplication: 𝛼(𝑨+𝑩)=𝛼𝑨+𝛼𝑩
Scalar Association of Scalar Multiplication: 𝛼(𝛽𝑨)=(𝛼𝛽)𝑨

Closure Law of Scalar Multiplication

The set 𝑆 of 𝑛-tuples is closed under multiplication by scalar because the multiplication of an element of the set by a scalar always produces another element in the set. That is 𝑨∊𝑆, 𝛼∊ℝ and 𝛼𝑨∊𝑆.

Let 𝑨=(𝐴₁,𝐴₂,⋯,𝐴_𝑛); 𝑨∊𝑆; 𝛼∊ℝ
    Let 𝑩=𝛼𝑨=𝛼(𝐴₁,𝐴₂,⋯,𝐴_𝑛). 
    ⇒𝑩=(𝛼𝐴₁,𝛼𝐴₂,⋯,𝛼𝐴_𝑛), by scalar multiplication property
    ⇒𝐵_𝑖=(𝛼𝐴_𝑖), where 𝑖=1,2,⋯,𝑛
    ∵ multiplication of real numbers is closed, ∴ all components of 𝑛-tuple, 𝐵_𝑖=(𝛼𝐴_𝑖) are real numbers
            ⇒𝑩=(𝛼𝐴₁,𝛼𝐴₂,⋯,𝛼𝐴_𝑛)=𝛼𝑨 is also in set 𝑆.
            ⇒𝛼𝑨 is closed. ∎

Scalar Identity of Scalar Multiplication

There only exists one unique scalar identity, 1, in ℝ such that the multiplication operation of any element in set 𝑆 by the scalar identity remains unchaged. In other words, the multiplication of the unique scalar identity, 1, as augand with any element in set 𝑆 is always equal to the element itself. That is 1𝑨=𝑨.

Let 𝑨=(𝐴₁,𝐴₂,⋯,𝐴_𝑛); 𝑨∊𝑆; 𝛼,𝛽∊ℝ
        𝛼𝑨=𝑨, by definition of scalar identity
    ⇒𝛼(𝐴₁,𝐴₂,⋯,𝐴_𝑛)=(𝐴₁,𝐴₂,⋯,𝐴_𝑛) 
    ⇒(𝛼𝐴₁,𝛼𝐴₂,⋯,𝛼𝐴_𝑛)=(𝐴₁,𝐴₂,⋯,𝐴_𝑛), by scalar multiplication property
    ⇒(𝛼𝐴_𝑖)=𝐴_𝑖, where 𝑖=1,2,⋯,𝑛
∵ 𝛼 and 𝐴₁ are real numbers, ∴ there exists only one unique real number solution, 𝛼=1, for any 𝐴₁
    ⇒(1𝐴_𝑖)=𝐴_𝑖, where 𝑖=1,2,⋯,𝑛
        ⇒(1𝐴₁,1𝐴₂,⋯,1𝐴_𝑛)=(𝐴₁,𝐴₂,⋯,𝐴_𝑛)
        ⇒1(𝐴₁,𝐴₂,⋯,𝐴_𝑛)=(𝐴₁,𝐴₂,⋯,𝐴_𝑛), by scalar multiplication property
        ⇒1𝑨=𝑨
        ∴ 𝛼𝑨=1𝑨=𝑨, where 𝛼=1 is the scalar identity
    ⇒1𝑨=𝑨, where 1 is the scalar identity. ∎

Vector Distribution Law of Scalar Multiplication

The product of the sum of two scalars with an 𝑛-tuple can be redistributed into the sum of the two products of each scalar with an 𝑛-tuple without changing the result. In other words, the scalar multiplication of an 𝑛-tuple by the sum of two scalars can be distributed as the sum of the scalar multiplication of an 𝑛-tuple by each scalar accordingly. That is (𝛼+𝛽)𝑨=𝛼𝑨+𝛽𝑨.

Let 𝑨=(𝐴₁,𝐴₂,⋯,𝐴_𝑛); 𝑨∊𝑆; 𝛼,𝛽∊ℝ.
   Let 𝑩=(𝛼+𝛽)𝑨=(𝛼+𝛽)(𝐴₁,𝐴₂,⋯,𝐴_𝑛)
   ⇒𝑩=((𝛼+𝛽)𝐴₁,(𝛼+𝛽)𝐴₂,⋯,(𝛼+𝛽)𝐴_𝑛), by scalar multiplication property
  ⇒𝐵_𝑖=((𝛼+𝛽)𝐴_𝑖), where 𝑖=1,2,⋯,𝑛
    ∵ real numbers is distributive, ∴ all components of 𝑛-tuple, ((𝛼+𝛽)𝐴_𝑖) can be rewrtiten as (𝛼𝐴_𝑖+𝛽𝐴_𝑖) without changing the result.    
⇒𝐵_𝑖=(𝛼𝐴_𝑖+𝛽𝐴_𝑖), where 𝑖=1,2,⋯,𝑛
⇒𝑩=(𝛼𝐴₁+𝛽𝐴₁,𝛼𝐴₂+𝛽𝐴₂,⋯,𝛼𝐴_𝑛+𝛽𝐴_𝑛)
        ⇒𝑩=(𝛼𝐴₁,𝛼𝐴₂,⋯,𝛼𝐴_𝑛)+(𝛽𝐴₁,𝛽𝐴₂,⋯,𝛽𝐴_𝑛), by addition property
        ⇒𝑩=𝛼(𝐴₁,𝐴₂,⋯,𝐴_𝑛)+𝛽(𝐴₁,𝐴₂,⋯,𝐴_𝑛), by scalar multiplication property
        ⇒𝑩=(𝛼+𝛽)𝑨=𝛼𝑨+𝛽𝑨
        ⇒(𝛼+𝛽)𝑨=𝛼𝑨+𝛽𝑨 is vector distributive. ∎

Scalar Distribution Law of Scalar Multiplication

The product of a scalar with the sum of two 𝑛-tuples can be redistributed into the sum of the two products of the scalar with each 𝑛-tuple without changing the result. In other words, the scalar multiplication of the sum of two 𝑛-tuples by a scalar can be distributed as the sum of the scalar multiplication of each 𝑛-tuple by the scalar accordingly. That is 𝛼(𝑨+𝑩)=𝛼𝑨+𝛼𝑩.

Let 𝑨=(𝐴₁,𝐴₂,⋯,𝐴_𝑛); 𝑩=(𝐵₁,𝐵₂,⋯,𝐵_𝑛); 𝑨,𝑩∊𝑆; 𝛼∊ℝ.
   Let 𝘾=𝛼(𝑨+𝑩)=𝛼((𝐴₁,𝐴₂,⋯,𝐴_𝑛)+(𝐵₁,𝐵₂,⋯,𝐵_𝑛))
   ⇒𝘾=𝛼(𝐴₁+𝐵₁,𝐴₂+𝐵₂,⋯,𝐴_𝑛+𝐵_𝑛), by addition property
   ⇒𝘾=(𝛼(𝐴₁+𝐵₁),𝛼(𝐴₂+𝐵₂),⋯,𝛼(𝐴_𝑛+𝐵_𝑛)), by scalar multiplication property
  ⇒𝐶_𝑖=(𝛼(𝐴_𝑖+𝐵_𝑖)), where 𝑖=1,2,⋯,𝑛
    ∵ real numbers is distributive, ∴ all components of 𝑛-tuple, (𝛼(𝐴_𝑖+𝐵_𝑖)) can be rewrtiten as (𝛼𝐴_𝑖+𝛼𝐵_𝑖) without changing the result.    
⇒𝐶_𝑖=(𝛼𝐴_𝑖+𝛼𝐵_𝑖), where 𝑖=1,2,⋯,𝑛
⇒𝘾=(𝛼𝐴₁+𝛼𝐵₁,𝛼𝐴₂+𝛼𝐵₂,⋯,𝛼𝐴_𝑛+𝛼𝐵_𝑛)
        ⇒𝘾=(𝛼𝐴₁,𝛼𝐴₂,⋯,𝛼𝐴_𝑛)+(𝛼𝐵₁,𝛼𝐵₂,⋯,𝛼𝐵_𝑛), by addition property
        ⇒𝘾=𝛼(𝐴₁,𝐴₂,⋯,𝐴_𝑛)+𝛼(𝐵₁,𝐵₂,⋯,𝐵_𝑛), by scalar multiplication property
        ⇒𝘾=𝛼(𝑨+𝑩)=𝛼𝑨+𝛼𝑩
        ⇒𝛼(𝑨+𝑩)=𝛼𝑨+𝛼𝑩 is scalar distributive. ∎

Scalar Association of Scalar Multiplication

The product of a scalar with the product of another scalar with an 𝑛-tuple in set 𝑆 can be re-associated into the product of the product of two scalar with an 𝑛-tuple in set 𝑆 without changing the result. That is 𝛼(𝛽𝑨)=(𝛼𝛽)𝑨.

Let 𝑨=(𝐴₁,𝐴₂,⋯,𝐴_𝑛); 𝑨∊𝑆; 𝛼,𝛽∊ℝ.
Let 𝑩=𝛼(𝛽𝑨)=𝛼(𝛽(𝐴₁,𝐴₂,⋯,𝐴_𝑛))
⇒𝑩=𝛼(𝛽𝐴₁,𝛽𝐴₂,⋯,𝛽𝐴_𝑛), by scalar multiplication property.
⇒𝑩=(𝛼(𝛽𝐴₁),𝛼(𝛽𝐴₂),⋯,𝛼(𝛽𝐴_𝑛)), by scalar multiplication property.
        ⇒𝐵_𝑖=(𝛼(𝛽𝐴_𝑖)), where 𝑖=1,2,⋯,𝑛
        ∵ real numbers is associative, ∴ all components of 𝑛-tuple, (𝛼(𝛽𝐴_𝑖)) can be rewrtiten as ((𝛼𝛽)𝐴_𝑖) without changing the result.
        ⇒𝐵_𝑖=((𝛼𝛽)𝐴_𝑖), where 𝑖=1,2,⋯,𝑛
        ⇒𝑩=((𝛼𝛽)𝐴₁,(𝛼𝛽)𝐴₂,⋯,(𝛼𝛽)𝐴_𝑛)
        ⇒𝑩=(𝛼𝛽)(𝐴₁,𝐴₂,⋯,𝐴_𝑛), by scalar multiplication property.
        ⇒𝑩=𝛼(𝛽𝑨)=(𝛼𝛽)𝑨
        ⇒𝛼(𝛽𝑨)=(𝛼𝛽)𝑨 is scalar associative. ∎

Fundamental Algebraic Laws for Scalar Multiplication

Fundamental Algebraic Laws for Scalar Multiplication
    Closure Law of Scalar Multiplication : If 𝛼 is element of ℝ and 𝑨 is elements of 𝑆, then 𝛼𝑨 is also element of 𝑆.
        Scalar Identity of Scalar Multiplication : There is only one unique real number, 1, such that 1𝑨=𝑨.
        Vector Distribution Law of Scalar Multiplication : The scalar multiplication of the sum of any two real numbers over an 𝑛-tuple in 𝑆 is distributive, that is (𝛼+𝛽)𝑨=𝛼𝑨+𝛽𝑨
        Scalar Distribution Law of Scalar Multiplication : The scalar multiplication of a real number over the sum of any two 𝑛-tuples in 𝑆 is distributive, that is 𝛼(𝑨+𝑩)=𝛼𝑨+𝛼𝑩
        Scalar Association of Scalar Multiplication : The scalar multiplication of a real number with the scalar multiplication of a real number with an 𝑛-tuple is associative, that is  𝛼(𝛽𝑨)=(𝛼𝛽)𝑨. ∎

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