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In summary, a vector space is a mathematical structure consisting of a set of objects called vectors and operations of addition and scalar multiplication. It follows a set of rules, such as closure and associativity, and has properties such as a zero vector and additive inverses. A vector is an element within a vector space. Examples of vector spaces include 2D and 3D vectors, polynomials, and continuous functions. Vector space is important in linear algebra because it allows for organized manipulation of mathematical objects and serves as the foundation for other crucial concepts.

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Mark44

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Miike012

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but e = (0,0) is not of the form (1,x)

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SammyS

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Miike012 said:but e = (0,0) is not of the form (1,x)

But for this vector space,

What is

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Miike012

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is it (1,0)?

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SammyS

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Yes, that'sMiike012 said:is it (1,0)?

A vector space is a mathematical structure that consists of a set of objects called vectors, along with operations of addition and scalar multiplication that allow for the manipulation and combination of these vectors. This structure follows a set of axioms or rules, which ensure that the operations are well-defined and behave in a consistent manner.

A vector space must satisfy the following properties: closure under addition and scalar multiplication, associativity and commutativity of addition, existence of a zero vector, existence of additive inverses, and distributivity of scalar multiplication over addition.

A vector is an element or object within a vector space. A vector space, on the other hand, is a mathematical structure that consists of a set of vectors and operations. Think of a vector space as a container that holds vectors, and a vector as one of the items inside that container.

Common examples of vector spaces include the set of all 2D or 3D vectors, the set of all n-dimensional vectors, the set of all polynomials with real coefficients, and the set of all continuous functions on a given interval.

Vector space is a fundamental concept in linear algebra because it provides a way to represent and manipulate mathematical objects in a systematic and organized manner. It also serves as the foundation for many other important concepts, such as linear transformations, bases, and eigenvalues, which are crucial in solving problems in various fields, including physics, engineering, and computer science.

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