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ContentGeneral Vector Space
General Vector SpaceA vector space is also called linear vector space and is the name used to describle a collection of objects called vectors. In other words, a vector space can be considered as a set and vectors are elements of the set. Euclidean vector space are the most common vector space for representing physical quantities which are usually visualized in form of an arrowlike vector. However, for vectors in 𝑛space, the form of vectors do not necessarily to be an arrowlike object. Besides, the form of element is also not limit to numeric numbers.Notations and ConventionsLet set 𝑆 or other capital letter be an 𝑛tuple vector space or 𝑛space. Elements of set 𝑆, 𝑛tuples, are usually represented by a bold face capital letter, e.g. 𝑨=(𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}), and elements of the 𝑛tuple 𝑨, i.e. (𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}) are called components of 𝑨. And for convenient, components of 𝑨 can be expressed as indexed elements with corresponding capital letter, i.e. 𝐴_{𝑖}, where 𝑖=1,2,⋯,𝑛. An ordered 𝑛tuple vector is always the element of an 𝑛tuple space.Real Vector SpaceReal vector space is one of the common vector space, for example ℝ^{2}, ℝ^{3} for 2, 3dimensional space. A vector space is named real vector space because components of all vector elements in the real vector space are real numbers. A real vector space itself is a collection of vector objects on which two standard operations, namely addition and scalar multipication, with typical properties are defined. In other words, 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.Addition and Scalar Multiplication OperationsThe addition and scalar multiplication operations is a rule set used to govern the available operations for real vector space.
Definition of Real Vector SpaceReal Vector SpaceLet the elements of set 𝑆 be ordered 𝑛tuple vectors of real numbers. Let 𝑨 be (𝐴_{1},𝐴_{2},⋯,𝐴_{𝑛}), 𝑩 be (𝐵_{1},𝐵_{2},⋯,𝐵_{𝑛}) where 𝐴_{𝑖} and 𝐵_{𝑖} are real numbers and 𝑖=1,2,⋯,𝑛. If elements of set 𝑆 satisfy the following operation properties:
Algebraic PropertiesThe addition and scale multiplication properties is only used to define the available operations for a real vector space. Two fundamental properties of vector elements are needed to be defined before defining the specific properties of binary operations of real vector space. The equal and zero properties are used to clarify the basic algebraic properties of vector elements in real vector space.
Definition of Algebraic PropertiesAlgebraic PropertiesThe two fundamental algebraic properties of 𝑛tuples in real vector space are:
©sideway ID: 200201402 Last Updated: 14/2/2020 Revision: 0 Ref: References
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