What characterizes a linearly independent set of vectors?

A linearly independent set of vectors is defined by the condition that the only way to form the zero vector as a linear combination of those vectors is by taking all the coefficients to be zero. This means that no vector in the set can be written as a linear combination of the others, which contributes to the uniqueness of their representation in the vector space.

When you consider linearly independent vectors, the only solution to the equation (c_1\mathbf{v_1} + c_2\mathbf{v_2} + ... + c_n\mathbf{v_n} = 0) must be (c_1 = c_2 = ... = c_n = 0). If any non-zero coefficients exist, it would indicate that at least one vector can be represented by a combination of the others, contradicting the definition of linear independence.

This principle is fundamental in linear algebra, as linear independence is crucial for forming a basis of a vector space. A set that meets this criterion ensures that the space spanned by these vectors does not have any redundancy, providing a complete and minimal representation of the space.