Linear Independence

A set of vectors is linearly independent if none of them can be written as a linear combination of the others, meaning no vector is redundant.

Formally, $v_1, \dots, v_k$ are linearly independent when the only way to satisfy $c_1 v_1 + \dots + c_k v_k = \mathbf{0}$ is with all coefficients zero ($c_1 = \dots = c_k = 0$).

If any nontrivial combination (not all zero) gives $\mathbf{0}$, they are linearly dependent, and one vector can be expressed in terms of the others.

Testing Independence

Checking independence is a homogeneous linear system: $c_1 v_1 + \dots + c_k v_k = \mathbf{0}$. The vectors are independent when this system has only the trivial solution, meaning the matrix formed by the vectors as columns has full column rank (a pivot in every column).

Common pitfall: Thinking independence just means no vector is a multiple of another. That only works for two vectors! With three or more, a set can be dependent even if no vector is a multiple of any other—for instance, if the third vector is the sum of the first two.

Independence means no wasted vectors. Combined with spanning, it forms the foundation for a basis in the next lesson.