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ContentProperties of Algebraic Operations
Properties of Algebraic OperationsBy 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 𝑛tuples for addition and scalar multiplication operations used in real vector space.Algebraic Laws for AdditionLet set 𝑆 be an 𝑛Tuple Vector Space. The 𝑛tuples of set 𝑆 also satisfy some fundamental algebraic laws for the addition operation. That is
Closure Law of AdditionThe set 𝑆 of 𝑛tuples 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=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}); 𝑩=(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛}); 𝑨,𝑩∊𝑆
Let 𝑪=𝑨+𝑩=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛}).
⇒𝑪=𝑨+𝑩=(𝐴_{1}+𝐵_{1},𝐴_{2}+𝐵_{2},⋯,𝐴_{𝑛}+𝐵_{𝑛}), by addition property
⇒𝐶_{𝑖}=(𝐴_{𝑖}+𝐵_{𝑖}), where 𝑖=1,2,⋯,𝑛
∵ addition of real numbers is closed, ∴ all components of 𝑛tuple, 𝐶_{𝑖}=𝐴_{𝑖}+𝐵_{𝑖} are real numbers
⇒𝑪=(𝐴_{1}+𝐵_{1},𝐴_{2}+𝐵_{2},⋯,𝐴_{𝑛}+𝐵_{𝑛})=𝑨+𝑩 is also in set 𝑆.
⇒𝑨+𝑩 is closed. ∎
Commutative Law of AdditionThe addition of any two 𝑛tuples 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=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}); 𝑩=(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛}); 𝑨,𝑩∊𝑆.
Let 𝑪=𝑨+𝑩=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛}).
⇒𝑪=(𝐴_{1}+𝐵_{1},𝐴_{2}+𝐵_{2},⋯,𝐴_{𝑛}+𝐵_{𝑛}), by addition property
⇒𝐶_{𝑖}=(𝐴_{𝑖}+𝐵_{𝑖}), where 𝑖=1,2,⋯,𝑛
∵ addition of real numbers is commutative, ∴ all components of 𝑛tuple, 𝐴_{𝑖}+𝐵_{𝑖} can be swapped to 𝐵_{𝑖}+𝐴_{𝑖}
⇒𝐶_{𝑖}=(𝐴_{𝑖}+𝐵_{𝑖})=(𝐵_{𝑖}+𝐴_{𝑖}), where 𝑖=1,2,⋯,𝑛
⇒𝑪=𝑨+𝑩=(𝐵_{1}+𝐴_{1},𝐵_{2}+𝐴_{2},⋯,𝐵_{𝑛}+𝐴_{𝑛})
⇒𝑪=𝑨+𝑩=(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛})+(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}), by addition property
⇒𝑪=𝑨+𝑩=𝑩+𝑨
⇒𝑨+𝑩=𝑩+𝑨 is commutative. ∎
Associative Law of AdditionThe addition of any three 𝑛tuples 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=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}); 𝑩=(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛}); 𝑪=(𝐶_{1},𝐶_{2},⋯,𝐶_{𝑛}); 𝑨,𝑩,𝑪∊𝑆.
Let 𝑫=(𝑨+𝑩)+𝑪=((𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛}))+(𝐶_{1},𝐶_{2},⋯,𝐶_{𝑛})
⇒ 𝑫=((𝐴_{1}+𝐵_{1},𝐴_{2}+𝐵_{2},⋯,𝐴_{𝑛}+𝐵_{𝑛}))+(𝐶_{1},𝐶_{2},⋯,𝐶_{𝑛}), by addition property
⇒𝑫=((𝐴_{1}+𝐵_{1})+𝐶_{1},(𝐴_{2}+𝐵_{2})+𝐶_{2},⋯,(𝐴_{𝑛}+𝐵_{𝑛})+𝐶_{𝑛}), by addition property
⇒𝐷_{𝑖}=(𝐴_{𝑖}+𝐵_{𝑖})+𝐶_{𝑖}, where 𝑖=1,2,⋯,𝑛
∵ addition of real numbers is associative, ∴ all components of 𝑛tuple, (𝐴_{𝑖}+𝐵_{𝑖})+𝐶_{𝑖} can be rewrtiten as 𝐴_{𝑖}+(𝐵_{𝑖}+𝐶_{𝑖}) without changing the summation result.
⇒𝐷_{𝑖}=(𝐴_{𝑖}+𝐵_{𝑖})+𝐶_{𝑖}=𝐴_{𝑖}+(𝐵_{𝑖}+𝐶_{𝑖}), where 𝑖=1,2,⋯,𝑛
⇒𝑫=(𝑨+𝑩)+𝑪=(𝐴_{1}+(𝐵_{1}+𝐶_{1}),𝐴_{2}+(𝐵_{2}+𝐶_{2}),⋯,𝐴_{𝑛}+(𝐵_{𝑛}+𝐶_{𝑛}))
⇒𝑫=(𝑨+𝑩)+𝑪=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(𝐵_{1}+𝐶_{1},𝐵_{2}+𝐶_{2},⋯,𝐵_{𝑛}+𝐶_{𝑛}), by addition property
⇒𝑫=(𝑨+𝑩)+𝑪=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+((𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛})+(𝐶_{1},𝐶_{2},⋯,𝐶_{𝑛})), by addition property
⇒𝑫=(𝑨+𝑩)+𝑪=𝑨+(𝑩+𝑪)
⇒(𝑨+𝑩)+𝑪=𝑨+(𝑩+𝑪) is associative. ∎
Additive IdentityThere 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=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}); 𝑿=(𝑋_{1},𝑋_{2},⋯,𝑋_{𝑛}); 𝑨,𝑿∊𝑆.
Let 𝑿 be an additive identity
𝑨+𝑿=𝑨, by definition of additive identity
⇒(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(𝑋_{1},𝑋_{2},⋯,𝑋_{𝑛})=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})
⇒(𝐴_{1}+𝑋_{1},𝐴_{2}+𝑋_{2},⋯,𝐴_{𝑛}+𝑋_{𝑛})=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}), by addition property
⇒(𝐴_{𝑖}+𝑋_{𝑖})=𝐴_{𝑖}, where 𝑖=1,2,⋯,𝑛
∵ 𝐴_{𝑖} and 𝑋_{𝑖} are real numbers, ∴ there exists only one unique real number solution, 𝑋_{𝑖}=0, for any 𝐴_{𝑖}
⇒(𝐴_{𝑖}+𝑋_{𝑖})=𝐴_{𝑖}, where 𝑋_{𝑖}=0; 𝑖=1,2,⋯,𝑛
⇒(𝐴_{1}+0,𝐴_{2}+0,⋯,𝐴_{𝑛}+0)=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})
⇒(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(0,0,⋯,0)=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}), 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: 𝑛tuple 𝟎 is the unique additive identity of 𝑛tuple vector space. ∎
Additive InverseFor 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 𝑛tuple 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=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}); 𝑿=(𝑋_{1},𝑋_{2},⋯,𝑋_{𝑛}); 𝒀=(𝑌_{1},𝑌_{2},⋯,𝑌_{𝑛}); 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.
⇒(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(𝑋_{1},𝑋_{2},⋯,𝑋_{𝑛})=(0,0,⋯,0)
⇒(𝐴_{1}+𝑋_{1},𝐴_{2}+𝑋_{2},⋯,𝐴_{𝑛}+𝑋_{𝑛})=(0,0,⋯,0), by addition property
⇒(𝐴_{𝑖}+𝑋_{𝑖})=0, where 𝑖=1,2,⋯,𝑛
∵ 𝐴_{𝑖}, 𝑋_{𝑖}, and 0 are real numbers, ∴ there exists only one unique real number solution, 𝑋_{𝑖}=(−𝐴_{𝑖}), for any 𝐴_{𝑖}. That is the negation of a real number.
⇒(𝐴_{𝑖}+𝑋_{𝑖})=0, where 𝑋_{𝑖}=(−𝐴_{𝑖}); 𝑖=1,2,⋯,𝑛
⇒(𝐴_{1}+(−𝐴_{1}),𝐴_{2}+(−𝐴_{2}),⋯,𝐴_{𝑛}+(−𝐴_{𝑛}))=(0,0,⋯,0)
⇒(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+((−𝐴_{1}),(−𝐴_{2}),⋯,(−𝐴_{𝑛}))=(0,0,⋯,0), by addition property
⇒(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(−(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}))=(0,0,⋯,0), by scalar multiplication property
⇒𝑨+(−𝑨)=𝟎
⇒𝑿=−(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})=(−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=−(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})=(−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 𝑛tuple vector 𝑨 is equal to the negation of the corresponding 𝑛tuple vector, i.e. −𝑨.
⇒𝑿=𝑋_{𝑖}=(−𝐴_{𝑖})=−(𝐴_{𝑖})=−𝑨, where 𝑖=1,2,⋯,𝑛, by scalar multiplication property of 𝑛tuple 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 𝑛tuple 𝑨 in set 𝑆, 𝑛tuple (−𝑨), the negation, is the unique addition inverse in 𝑛tuple vector space. ∎
Fundamental Algebraic Laws for AdditionFundamental Algebraic Laws for Addition
Inverse Addition OperationThe 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 𝑛tuple is equal to the subtraction of that original 𝑛tuple. That isLet 𝑨=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}); 𝑩=(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛}); 𝑨,𝑩∊𝑆
Let 𝑪=𝑨+(−𝑩)=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(−(𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛})).
Let 𝑪=𝑨+(−𝑩)=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛})+(−𝐵_{1},−𝐵_{2},⋯,−𝐵_{𝑛}), by scalar multiplication property.
⇒𝑪=𝑨+(−𝑩)=(𝐴_{1}+(−𝐵_{1}),𝐴_{2}+(−𝐵_{2}),⋯,𝐴_{𝑛}+(−𝐵_{𝑛})), by addition property
⇒𝑪=𝑨+(−𝑩)=(𝐴_{1}−𝐵_{1},𝐴_{2}−𝐵_{2},⋯,𝐴_{𝑛}−𝐵_{𝑛}), by real number property
⇒𝑪=𝑨+(−𝑩)=𝑨−𝑩, consider '−' as an inverse addition operation, called subtraction operation
Definition of SubtractionDefinition 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.
©sideway ID: 200201802 Last Updated: 18/2/2020 Revision: 0 Ref: References
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