jmm said:
)A vector space is a mathematical structure that consists of a set of objects, called vectors, and a set of operations, such as addition and scalar multiplication, that can be performed on those vectors. The properties of a vector space include closure under addition and scalar multiplication, as well as the existence of a zero vector and additive inverses.
The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it is the set of all vectors that can be created by multiplying each vector by a scalar and adding them together. The span of a set of vectors is always a subspace of the vector space in which the vectors reside.
To prove that a set of vectors spans a vector space, you must show that every vector in the vector space can be expressed as a linear combination of the given set of vectors. This can be done by writing out the general form of a linear combination and then setting it equal to a specific vector in the vector space. If you can solve for the coefficients of the linear combination, then the vector is in the span of the given set of vectors.
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 the set. In other words, no vector in the set is redundant and each vector brings something new to the span of the set. A linearly independent set of vectors is important in proving that a set of vectors spans a vector space.
To prove that a set of vectors is linearly independent, you must show that the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are scalars and v1, v2, ..., vn are the vectors in the set, is when all of the coefficients are equal to 0. This can be done by setting up a system of equations and using techniques such as Gaussian elimination to solve for the coefficients.