Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of each other. In other words, no vector in the set can be expressed as a linear combination of the other vectors in the set.
A set of vectors is linearly independent if and only if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is when all the coefficients c1, c2, ..., cn are equal to 0. This means that none of the vectors can be written as a linear combination of the others.
Linear independence is an important concept in linear algebra because it allows us to understand the structure and behavior of vector spaces. It also helps us to solve systems of linear equations and to determine the dimension of a vector space.
No, a set of vectors cannot be both linearly independent and linearly dependent. They are mutually exclusive concepts. A set of vectors can only be either linearly independent or linearly dependent.
If a set of vectors is linearly independent, then the span of those vectors will be equal to the number of vectors in the set. On the other hand, if a set of vectors is linearly dependent, then the span of those vectors will be less than the number of vectors in the set.