A set of vectors $\{\mathbf v_1, \dots, \mathbf v_n\}$​ is *linearly independent* if the only solution to their linear combination equaling the zero vector is when all scalar coefficients are zero: $ c_1 \mathbf v_1 + \cdots + c_n \mathbf v_n = 0, \quad \text{where} \quad c_1=\dots = c_n=0$ Linear independence means that no vector in the set can be written as a linear combination of the others. If even one vector can be expressed in terms of the others, the set is *linearly dependent*. **Unique Representation:** In a linearly independent set, each vector adds a new, unique direction or dimension to the vector space. No redundancy exists. **Matrix Interpretation:** - Linear independence can be evaluated row-wise or column-wise in a matrix, and both yield the same result. - A set of column vectors $\{\mathbf v_1, \dots, \mathbf v_n \}$ is linearly independent if the matrix formed by these vectors as columns has full rank.