A vector space is a mathematical structure that consists of a set of vectors, which can be added together and multiplied by scalars (usually real numbers) to produce new vectors. It satisfies a set of axioms, including closure under addition and scalar multiplication, and the existence of a zero vector and additive inverses.
The basis of a vector space is a set of linearly independent vectors that span the entire vector space. This means that any vector in the space can be uniquely expressed as a linear combination of the basis vectors. In other words, the basis forms a "coordinate system" for the vector space.
In order for a set of vectors to form a basis for a vector space, they must satisfy two conditions: they must be linearly independent, meaning that no vector in the set can be expressed as a linear combination of the others, and they must span the entire vector space, meaning that every vector in the space can be expressed as a linear combination of the basis vectors.
Yes, a vector space can have multiple bases. This is because there are often many different sets of linearly independent vectors that can span a given vector space. However, all bases for a particular vector space will have the same number of basis vectors, known as the dimension of the vector space.
A basis is a set of linearly independent vectors that span a vector space. This means that the basis vectors are the minimum number of vectors necessary to express every other vector in the space. If a set of vectors is not linearly independent, it cannot form a basis for the vector space.