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A vector space is a concept from mathematics. A vector space is a collection of objects called vectors. Two operations are defined: addition of two vectors and multiplication of a vector with a scalar (multiplication results in scaling). More formally, a vector space is a special combination of a group and a field. The elements of the group are called vectors and the elements of the field are called scalars.
These "vectors" do not have to be vectors in the simplest sense. For example, they could be functions, matrices or simply numbers. So long as they obey the axioms of a vector space, we can think of them as vectors and the theorems of linear algebra will apply to them.
Often, for example in Euclidean space, a vector can be represented graphically with an arrow that has a tail and a head. We normally then think of addition as the tail of one vector being placed at the same point as the head of the other vector. The sum vector is the one whose tail is the tail of the first vector and whose head is the head of the second. Multiplication means that one vector is made bigger or smaller.
There are some combinations of vectors that are special. A minimum set of vectors that, through some combination of addition and multiplication, can reach any point in the vector space is called a basis of that vector space. It is true that every vector space has a basis. It is also true that all bases of any one vector space have the same number of vectors in them. This is called the dimension theorem. We can then define the dimension of a vector space to be the size of its basis.