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,⋯,𝐴𝑛); 𝑨∊𝑆; 𝛼∊ℝ Let 𝑩=𝛼𝑨=𝛼(𝐴1,𝐴2,⋯,𝐴𝑛). ⇒𝑩=(𝛼𝐴1,𝛼𝐴2,⋯,𝛼𝐴𝑛), by scalar multiplication property ⇒𝐵𝑖=(𝛼𝐴𝑖), where 𝑖=1,2,⋯,𝑛 ∵ multiplication of real numbers is closed, ∴ all components of 𝑛-tuple, 𝐵𝑖=(𝛼𝐴𝑖) are real numbers ⇒𝑩=(𝛼𝐴1,𝛼𝐴2,⋯,𝛼𝐴𝑛)=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,⋯,𝐴𝑛); 𝑨∊𝑆; 𝛼,𝛽∊ℝ 𝛼𝑨=𝑨, by definition of scalar identity ⇒𝛼(𝐴1,𝐴2,⋯,𝐴𝑛)=(𝐴1,𝐴2,⋯,𝐴𝑛) ⇒(𝛼𝐴1,𝛼𝐴2,⋯,𝛼𝐴𝑛)=(𝐴1,𝐴2,⋯,𝐴𝑛), by scalar multiplication property ⇒(𝛼𝐴𝑖)=𝐴𝑖, where 𝑖=1,2,⋯,𝑛 ∵ 𝛼 and 𝐴1 are real numbers, ∴ there exists only one unique real number solution, 𝛼=1, for any 𝐴1 ⇒(1𝐴𝑖)=𝐴𝑖, where 𝑖=1,2,⋯,𝑛 ⇒(1𝐴1,1𝐴2,⋯,1𝐴𝑛)=(𝐴1,𝐴2,⋯,𝐴𝑛) ⇒1(𝐴1,𝐴2,⋯,𝐴𝑛)=(𝐴1,𝐴2,⋯,𝐴𝑛), 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,⋯,𝐴𝑛); 𝑨∊𝑆; 𝛼,𝛽∊ℝ. Let 𝑩=(𝛼+𝛽)𝑨=(𝛼+𝛽)(𝐴1,𝐴2,⋯,𝐴𝑛) ⇒𝑩=((𝛼+𝛽)𝐴1,(𝛼+𝛽)𝐴2,⋯,(𝛼+𝛽)𝐴𝑛), by scalar multiplication property ⇒𝐵𝑖=((𝛼+𝛽)𝐴𝑖), where 𝑖=1,2,⋯,𝑛 ∵ real numbers is distributive, ∴ all components of 𝑛-tuple, ((𝛼+𝛽)𝐴𝑖) can be rewrtiten as (𝛼𝐴𝑖+𝛽𝐴𝑖) without changing the result. ⇒𝐵𝑖=(𝛼𝐴𝑖+𝛽𝐴𝑖), where 𝑖=1,2,⋯,𝑛 ⇒𝑩=(𝛼𝐴1+𝛽𝐴1,𝛼𝐴2+𝛽𝐴2,⋯,𝛼𝐴𝑛+𝛽𝐴𝑛) ⇒𝑩=(𝛼𝐴1,𝛼𝐴2,⋯,𝛼𝐴𝑛)+(𝛽𝐴1,𝛽𝐴2,⋯,𝛽𝐴𝑛), by addition property ⇒𝑩=𝛼(𝐴1,𝐴2,⋯,𝐴𝑛)+𝛽(𝐴1,𝐴2,⋯,𝐴𝑛), 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,⋯,𝐴𝑛); 𝑩=(𝐵1,𝐵2,⋯,𝐵𝑛); 𝑨,𝑩∊𝑆; 𝛼∊ℝ. Let 𝘾=𝛼(𝑨+𝑩)=𝛼((𝐴1,𝐴2,⋯,𝐴𝑛)+(𝐵1,𝐵2,⋯,𝐵𝑛)) ⇒𝘾=𝛼(𝐴1+𝐵1,𝐴2+𝐵2,⋯,𝐴𝑛+𝐵𝑛), by addition property ⇒𝘾=(𝛼(𝐴1+𝐵1),𝛼(𝐴2+𝐵2),⋯,𝛼(𝐴𝑛+𝐵𝑛)), by scalar multiplication property ⇒𝐶𝑖=(𝛼(𝐴𝑖+𝐵𝑖)), where 𝑖=1,2,⋯,𝑛 ∵ real numbers is distributive, ∴ all components of 𝑛-tuple, (𝛼(𝐴𝑖+𝐵𝑖)) can be rewrtiten as (𝛼𝐴𝑖+𝛼𝐵𝑖) without changing the result. ⇒𝐶𝑖=(𝛼𝐴𝑖+𝛼𝐵𝑖), where 𝑖=1,2,⋯,𝑛 ⇒𝘾=(𝛼𝐴1+𝛼𝐵1,𝛼𝐴2+𝛼𝐵2,⋯,𝛼𝐴𝑛+𝛼𝐵𝑛) ⇒𝘾=(𝛼𝐴1,𝛼𝐴2,⋯,𝛼𝐴𝑛)+(𝛼𝐵1,𝛼𝐵2,⋯,𝛼𝐵𝑛), by addition property ⇒𝘾=𝛼(𝐴1,𝐴2,⋯,𝐴𝑛)+𝛼(𝐵1,𝐵2,⋯,𝐵𝑛), 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,⋯,𝐴𝑛); 𝑨∊𝑆; 𝛼,𝛽∊ℝ. Let 𝑩=𝛼(𝛽𝑨)=𝛼(𝛽(𝐴1,𝐴2,⋯,𝐴𝑛)) ⇒𝑩=𝛼(𝛽𝐴1,𝛽𝐴2,⋯,𝛽𝐴𝑛), by scalar multiplication property. ⇒𝑩=(𝛼(𝛽𝐴1),𝛼(𝛽𝐴2),⋯,𝛼(𝛽𝐴𝑛)), by scalar multiplication property. ⇒𝐵𝑖=(𝛼(𝛽𝐴𝑖)), where 𝑖=1,2,⋯,𝑛 ∵ real numbers is associative, ∴ all components of 𝑛-tuple, (𝛼(𝛽𝐴𝑖)) can be rewrtiten as ((𝛼𝛽)𝐴𝑖) without changing the result. ⇒𝐵𝑖=((𝛼𝛽)𝐴𝑖), where 𝑖=1,2,⋯,𝑛 ⇒𝑩=((𝛼𝛽)𝐴1,(𝛼𝛽)𝐴2,⋯,(𝛼𝛽)𝐴𝑛) ⇒𝑩=(𝛼𝛽)(𝐴1,𝐴2,⋯,𝐴𝑛), 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. ∎