A vector space is a mathematical structure that consists of a set of vectors and a set of operations that can be performed on those vectors. These operations include addition and scalar multiplication, and they must follow certain rules in order for the set to be considered a vector space.
A subspace is a subset of a vector space that also satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and it contains the zero vector.
A set of vectors is linearly independent if none of the vectors in the set can be expressed as a linear combination of the other vectors. In other words, no vector in the set is redundant and adding or removing a vector would change the span of the set. This can be determined by setting up a system of equations and solving for the coefficients.
A basis is a set of linearly independent vectors that span the entire vector space, meaning that any vector in the vector space can be written as a linear combination of the basis vectors. A spanning set is a set of vectors that can generate the entire vector space, but they may not necessarily be linearly independent.
Yes, a vector space can have many different bases. In fact, any linearly independent set of vectors that spans the vector space can be considered a basis. However, all bases for a particular vector space must have the same number of vectors, which is known as the dimension of the vector space.