A subset is linearly independent if none of its vectors can be written as a linear combination of the other vectors in the subset. In other words, no vector in the subset is redundant and all the vectors are necessary to span the vector space.
To test if a subset is linearly independent, you can use the definition of linear independence by setting up a system of equations and solving for the coefficients. If there is a unique solution where all the coefficients are equal to 0, then the subset is linearly independent.
Linear independence and spanning are two different concepts. Linear independence refers to the relationship between vectors in a subset, while spanning refers to the ability of a subset to cover or reach every vector in a vector space. A subset can be linearly independent and not span the entire vector space, and vice versa.
Yes, a subset can be both linearly dependent and spanning. This means that there is at least one vector in the subset that can be written as a linear combination of the other vectors, but the subset is still able to cover or reach every vector in the vector space.
Basis vectors are a set of linearly independent vectors that span a vector space. This means that every vector in the vector space can be written as a unique linear combination of the basis vectors. Therefore, linear independence is a necessary condition for a set of vectors to be a basis.