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 Addition

Let set 𝑆 be an 𝑛-Tuple Vector Space. The 𝑛-tuples of set 𝑆 also satisfy some fundamental algebraic laws for the addition operation. That is

• Closure Law of Addition: If 𝑨,𝑩∊𝑆, then 𝑨+𝑩∊𝑆
• Commutative Law of Addition: 𝑨+𝑩=𝑩+𝑨
• Associative Law of Addition: (𝑨+𝑩)+𝑪=𝑨+(𝑩+𝑪)
• Additive Identity of Addition: zero 𝑛-tuple 𝟎∊𝑆:𝑨+𝟎=𝟎+𝑨=𝑨
• Additive Inverse of Addition: negation of any 𝑨, (−𝑨):𝑨+(−𝑨)=(−𝑨)+𝑨=𝟎

Closure Law of Addition

The set 𝑆 of 𝑛-tuples is closed under addition because the addition of any two elements of the set always produces another element in the set. That is 𝑨,𝑩∊𝑆 and 𝑨+𝑩∊𝑆. Let 𝑨=(𝐴1,𝐴2,⋯,𝐴𝑛); 𝑩=(𝐵1,𝐵2,⋯,𝐵𝑛); 𝑨,𝑩∊𝑆 Let 𝑪=𝑨+𝑩=(𝐴1,𝐴2,⋯,𝐴𝑛)+(𝐵1,𝐵2,⋯,𝐵𝑛). ⇒𝑪=𝑨+𝑩=(𝐴1+𝐵1,𝐴2+𝐵2,⋯,𝐴𝑛+𝐵𝑛), by addition property ⇒𝐶𝑖=(𝐴𝑖+𝐵𝑖), where 𝑖=1,2,⋯,𝑛 ∵ addition of real numbers is closed, ∴ all components of 𝑛-tuple, 𝐶𝑖=𝐴𝑖+𝐵𝑖 are real numbers ⇒𝑪=(𝐴1+𝐵1,𝐴2+𝐵2,⋯,𝐴𝑛+𝐵𝑛)=𝑨+𝑩 is also in set 𝑆. ⇒𝑨+𝑩 is closed. ∎

Commutative Law of Addition

The addition of any two 𝑛-tuples in set 𝑆 is commutative because the addition of any two elements of the set is irrespective of their order in the binary operation. In other words, the augend and addend of an addition operation can be swapped without changing the summation result of an addition operation. That is 𝑨+𝑩=𝑩+𝑨. Let 𝑨=(𝐴1,𝐴2,⋯,𝐴𝑛); 𝑩=(𝐵1,𝐵2,⋯,𝐵𝑛); 𝑨,𝑩∊𝑆. Let 𝑪=𝑨+𝑩=(𝐴1,𝐴2,⋯,𝐴𝑛)+(𝐵1,𝐵2,⋯,𝐵𝑛). ⇒𝑪=(𝐴1+𝐵1,𝐴2+𝐵2,⋯,𝐴𝑛+𝐵𝑛), by addition property ⇒𝐶𝑖=(𝐴𝑖+𝐵𝑖), where 𝑖=1,2,⋯,𝑛 ∵ addition of real numbers is commutative, ∴ all components of 𝑛-tuple, 𝐴𝑖+𝐵𝑖 can be swapped to 𝐵𝑖+𝐴𝑖 ⇒𝐶𝑖=(𝐴𝑖+𝐵𝑖)=(𝐵𝑖+𝐴𝑖), where 𝑖=1,2,⋯,𝑛 ⇒𝑪=𝑨+𝑩=(𝐵1+𝐴1,𝐵2+𝐴2,⋯,𝐵𝑛+𝐴𝑛) ⇒𝑪=𝑨+𝑩=(𝐵1,𝐵2,⋯,𝐵𝑛)+(𝐴1,𝐴2,⋯,𝐴𝑛), by addition property ⇒𝑪=𝑨+𝑩=𝑩+𝑨 ⇒𝑨+𝑩=𝑩+𝑨 is commutative. ∎

Associative Law of Addition

The addition of any three 𝑛-tuples in set 𝑆 is associative because an addition operation sequence for any three elements of the set is irrespective of the preforming order of binary operations provided that the order of the operands in the sequence is not changed. In other words, by keeping a row of operands and addition operation symbols unchanged, the summation result of the given row is not changed when the order of performing addition operation is changed. That is (𝑨+𝑩)+𝑪=𝑨+(𝑩+𝑪). Let 𝑨=(𝐴1,𝐴2,⋯,𝐴𝑛); 𝑩=(𝐵1,𝐵2,⋯,𝐵𝑛); 𝑪=(𝐶1,𝐶2,⋯,𝐶𝑛); 𝑨,𝑩,𝑪∊𝑆. Let 𝑫=(𝑨+𝑩)+𝑪=((𝐴1,𝐴2,⋯,𝐴𝑛)+(𝐵1,𝐵2,⋯,𝐵𝑛))+(𝐶1,𝐶2,⋯,𝐶𝑛) ⇒ 𝑫=((𝐴1+𝐵1,𝐴2+𝐵2,⋯,𝐴𝑛+𝐵𝑛))+(𝐶1,𝐶2,⋯,𝐶𝑛), by addition property ⇒𝑫=((𝐴1+𝐵1)+𝐶1,(𝐴2+𝐵2)+𝐶2,⋯,(𝐴𝑛+𝐵𝑛)+𝐶𝑛), by addition property ⇒𝐷𝑖=(𝐴𝑖+𝐵𝑖)+𝐶𝑖, where 𝑖=1,2,⋯,𝑛 ∵ addition of real numbers is associative, ∴ all components of 𝑛-tuple, (𝐴𝑖+𝐵𝑖)+𝐶𝑖 can be rewrtiten as 𝐴𝑖+(𝐵𝑖+𝐶𝑖) without changing the summation result. ⇒𝐷𝑖=(𝐴𝑖+𝐵𝑖)+𝐶𝑖=𝐴𝑖+(𝐵𝑖+𝐶𝑖), where 𝑖=1,2,⋯,𝑛 ⇒𝑫=(𝑨+𝑩)+𝑪=(𝐴1+(𝐵1+𝐶1),𝐴2+(𝐵2+𝐶2),⋯,𝐴𝑛+(𝐵𝑛+𝐶𝑛)) ⇒𝑫=(𝑨+𝑩)+𝑪=(𝐴1,𝐴2,⋯,𝐴𝑛)+(𝐵1+𝐶1,𝐵2+𝐶2,⋯,𝐵𝑛+𝐶𝑛), by addition property ⇒𝑫=(𝑨+𝑩)+𝑪=(𝐴1,𝐴2,⋯,𝐴𝑛)+((𝐵1,𝐵2,⋯,𝐵𝑛)+(𝐶1,𝐶2,⋯,𝐶𝑛)), by addition property ⇒𝑫=(𝑨+𝑩)+𝑪=𝑨+(𝑩+𝑪) ⇒(𝑨+𝑩)+𝑪=𝑨+(𝑩+𝑪) is associative. ∎

Additive Identity

There only exists one unique additive identity, 𝟎, in set 𝑆 such that the additon operation of any element in set 𝑆 and the additive identity in either order remains unchaged. In other words, the addition of the unique additive identity, 𝟎, as addend or augand with any element in set 𝑆 is always equal to the element itself. That is 𝑨+𝟎=𝟎+𝑨=𝑨. Let 𝑨=(𝐴1,𝐴2,⋯,𝐴𝑛); 𝑿=(𝑋1,𝑋2,⋯,𝑋𝑛); 𝑨,𝑿∊𝑆. Let 𝑿 be an additive identity 𝑨+𝑿=𝑨, by definition of additive identity ⇒(𝐴1,𝐴2,⋯,𝐴𝑛)+(𝑋1,𝑋2,⋯,𝑋𝑛)=(𝐴1,𝐴2,⋯,𝐴𝑛) ⇒(𝐴1+𝑋1,𝐴2+𝑋2,⋯,𝐴𝑛+𝑋𝑛)=(𝐴1,𝐴2,⋯,𝐴𝑛), by addition property ⇒(𝐴𝑖+𝑋𝑖)=𝐴𝑖, where 𝑖=1,2,⋯,𝑛 ∵ 𝐴𝑖 and 𝑋𝑖 are real numbers, ∴ there exists only one unique real number solution, 𝑋𝑖=0, for any 𝐴𝑖 ⇒(𝐴𝑖+𝑋𝑖)=𝐴𝑖, where 𝑋𝑖=0; 𝑖=1,2,⋯,𝑛 ⇒(𝐴1+0,𝐴2+0,⋯,𝐴𝑛+0)=(𝐴1,𝐴2,⋯,𝐴𝑛) ⇒(𝐴1,𝐴2,⋯,𝐴𝑛)+(0,0,⋯,0)=(𝐴1,𝐴2,⋯,𝐴𝑛), by addition property ⇒𝑨+𝟎=𝑨 ⇒𝑿=(0,0,⋯,0)=𝟎 is the unique element in set 𝑆 for 𝑨+𝑿=𝑨 ∴ 𝑨+𝑿=𝑨+𝟎=𝑨, where 𝟎 is the additive identity ⇒𝑿+𝑨=𝟎+𝑨=𝑨, by commutative law of addition ⇒𝑿=𝟎 is the unique element in set 𝑆 for 𝑿+𝑨=𝑨 ∴ 𝑿+𝑨=𝟎+𝑨=𝑨, where 𝟎 is the additive identity ⇒𝑨+𝟎=𝟎+𝑨=𝑨, by equal property ⇒𝑨+𝟎=𝟎+𝑨=𝑨: 𝑛-tuple 𝟎 is the unique additive identity of 𝑛-tuple vector space. ∎

Additive Inverse

For every element, 𝑨, in set 𝑆, there always exists one unique additive inverse element, −𝑨, in set 𝑆 such that adding the additive inverse of an element, −𝑨, to the element, 𝑨, itself always yields a 𝑛-tuple 𝟎, where 𝟎 is the additive identity of set 𝑆. In other words, the additive inverse element is the opposite or negation of an element in set 𝑆 so as to yield the additive identity of set 𝑆 in an addition operation. That is 𝑨+(−𝑨)=(−𝑨)+𝑨=𝟎. Let 𝑨=(𝐴1,𝐴2,⋯,𝐴𝑛); 𝑿=(𝑋1,𝑋2,⋯,𝑋𝑛); 𝒀=(𝑌1,𝑌2,⋯,𝑌𝑛); 𝟎=(0,0,⋯,0); 𝑨,𝑿,𝒀,𝟎∊𝑆 Let both 𝑿 and 𝒀 are additive inverses of 𝑨 Suppose 𝑨+𝑿=𝟎 and 𝑨+𝒀=𝟎, by definition of additive inverse. ⇒𝟎=(𝑨+𝑿)=(𝑨+𝒀), by equal property ∵ (𝑨+𝑿)=(𝑨+𝒀), ∴ Adding the addition inverse of 𝑨, as augend to both sides still maintain the equal identity. ⇒𝑿+(𝑨+𝑿)=𝑿+(𝑨+𝒀), try adding 𝑿 as augend to both sides ⇒(𝑿+𝑨)+𝑿=(𝑿+𝑨)+𝒀, by associative law of addition ∵ 𝑨+𝑿=𝑨+𝒀=𝟎, ∴ 𝑿+𝑨=𝒀+𝑨=𝟎, by commutative law of addition ⇒𝟎+𝑿=𝟎+𝒀, ∵ 𝑿+𝑨=𝒀+𝑨=𝟎 ⇒𝑿=𝒀, by addition identity (Other approach: Suppose 𝑿 and 𝒀 are additive inverses of 𝑨. Then 𝑿=𝑿+𝟎=𝑿+(𝑨+𝒀)=(𝑿+𝑨)+𝒀=𝟎+𝒀=𝒀.) ∴ the additive inverse of an element is unique. ⇒𝑨+𝑿=𝟎, by definition of additive inverse. ⇒(𝐴1,𝐴2,⋯,𝐴𝑛)+(𝑋1,𝑋2,⋯,𝑋𝑛)=(0,0,⋯,0) ⇒(𝐴1+𝑋1,𝐴2+𝑋2,⋯,𝐴𝑛+𝑋𝑛)=(0,0,⋯,0), by addition property ⇒(𝐴𝑖+𝑋𝑖)=0, where 𝑖=1,2,⋯,𝑛 ∵ 𝐴𝑖, 𝑋𝑖, and 0 are real numbers, ∴ there exists only one unique real number solution, 𝑋𝑖=(−𝐴𝑖), for any 𝐴𝑖. That is the negation of a real number. ⇒(𝐴𝑖+𝑋𝑖)=0, where 𝑋𝑖=(−𝐴𝑖); 𝑖=1,2,⋯,𝑛 ⇒(𝐴1+(−𝐴1),𝐴2+(−𝐴2),⋯,𝐴𝑛+(−𝐴𝑛))=(0,0,⋯,0) ⇒(𝐴1,𝐴2,⋯,𝐴𝑛)+((−𝐴1),(−𝐴2),⋯,(−𝐴𝑛))=(0,0,⋯,0), by addition property ⇒(𝐴1,𝐴2,⋯,𝐴𝑛)+(−(𝐴1,𝐴2,⋯,𝐴𝑛))=(0,0,⋯,0), by scalar multiplication property ⇒𝑨+(−𝑨)=𝟎 ⇒𝑿=−(𝐴1,𝐴2,⋯,𝐴𝑛)=(−𝑨) is the unique element in set 𝑆 for 𝑨+𝑿=𝟎 ∴ 𝑨+𝑿=𝑨+(−𝑨)=𝟎, where (−𝑨) is the unique additive inverse of 𝑨 ⇒𝑿+𝑨=(−𝑨)+𝑨=𝟎, by commutative law of addition ⇒𝑿=−(𝐴1,𝐴2,⋯,𝐴𝑛)=(−𝑨) is the unique element in set 𝑆 for 𝑿+𝑨=𝟎 ∴ 𝑿+𝑨=(−𝑨)+𝑨=𝟎, where (−𝑨) is the unique additive inverse of 𝑨 ⇒𝑨+(−𝑨)=(−𝑨)+𝑨=𝟎, by equal property ⇒𝑨+(−𝑨)=(−𝑨)+𝑨=𝟎: the addition inverse of any 𝑛-tuple vector 𝑨 is equal to the negation of the corresponding 𝑛-tuple vector, i.e. −𝑨. ⇒𝑿=𝑋𝑖=(−𝐴𝑖)=−(𝐴𝑖)=−𝑨, where 𝑖=1,2,⋯,𝑛, by scalar multiplication property of 𝑛-tuple space ⇒𝑿=(−𝑨) is the unique element in set 𝑆 for 𝑨+𝑿=𝟎 ⇒ 𝑨+(−𝑨)=𝟎 ⇒𝑨+(−𝑨)=(−𝑨)+𝑨=𝟎, by commutative law of addition ⇒𝑿=(−𝑨) is the unique element in set 𝑆 for 𝑿+𝑨=𝟎 ⇒𝑨+(−𝑨)=(−𝑨)+𝑨=𝟎, by equal property ⇒𝑨+(−𝑨)=(−𝑨)+𝑨=𝟎: for each 𝑛-tuple 𝑨 in set 𝑆, 𝑛-tuple (−𝑨), the negation, is the unique addition inverse in 𝑛-tuple vector space. ∎

Fundamental Algebraic Laws for Addition

Fundamental Algebraic Laws for Addition Closure Law : If 𝑨 and 𝑩 are elements of 𝑆, then 𝑪=𝑨+𝑩 is also element of 𝑆. Commutative law of addition : 𝑨+𝑩=𝑩+𝑨. Associative law of addition : (𝑨+𝑩)+𝑪=𝑨+(𝑩+𝑪) Additive Identity of addition : The 𝑛-tuple 𝟎 is the unique element of 𝑆 with the property 𝑨+𝟎=𝟎+𝑨=𝑨 Additive inverse of addition : Corresponding to each 𝑨 in 𝑆, −𝑨 is the unique element such that 𝑨+(−𝑨)=(−𝑨)+𝑨=𝟎. ∎

Inverse Addition Operation

The negative property of additive inverse can be used as the subtraction concept to define the subtraction operation. In other words, the adding of additive inverse of an 𝑛-tuple is equal to the subtraction of that original 𝑛-tuple. That is Let 𝑨=(𝐴1,𝐴2,⋯,𝐴𝑛); 𝑩=(𝐵1,𝐵2,⋯,𝐵𝑛); 𝑨,𝑩∊𝑆 Let 𝑪=𝑨+(−𝑩)=(𝐴1,𝐴2,⋯,𝐴𝑛)+(−(𝐵1,𝐵2,⋯,𝐵𝑛)). Let 𝑪=𝑨+(−𝑩)=(𝐴1,𝐴2,⋯,𝐴𝑛)+(−𝐵1,−𝐵2,⋯,−𝐵𝑛), by scalar multiplication property. ⇒𝑪=𝑨+(−𝑩)=(𝐴1+(−𝐵1),𝐴2+(−𝐵2),⋯,𝐴𝑛+(−𝐵𝑛)), by addition property ⇒𝑪=𝑨+(−𝑩)=(𝐴1−𝐵1,𝐴2−𝐵2,⋯,𝐴𝑛−𝐵𝑛), by real number property ⇒𝑪=𝑨+(−𝑩)=𝑨−𝑩, consider '−' as an inverse addition operation, called subtraction operation

Definition of Subtraction

Definition of Subtraction<A subtraction operation is defined as the inverse operation to that of the type addition 𝑨−𝑩=𝑨+(−𝑩), inverse operation through the adding of additive inverse.