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Given a Field F a vector space satisfies a few axioms to be decreed as a vector space. Essentially it must satisfy the properties of commutativity, and associativety and under addition of elements of the vector space, it must also contain an identity vector and an inverse additively in the vector space as well as have a . The Field F comes in where we need it that scalar multiplication satisfies distributive properties, associative, and must be such that 1x = x where x is an element of the vector space and 1 is the multiplicative identity of Field F.
vector space means a space containing some vectors and if v1,v2 in those vector then c*v1+c*v2 lie in those vector/the vector space.c,d are all real number.