Linear Independence Theorem

The Linear Independence Theorem states that a set of vectors is linearly independent if no vector in the set can be written as a combination of the others. In practical terms, this means each vector adds a new, unique direction or dimension to the space; none are redundant. If vectors are linearly independent, they form a minimal basis for a space, capturing all its directions without overlap. Conversely, if one vector can be formed from others, they are linearly dependent, indicating some vectors are unnecessary for describing the whole space.