Linear (in)dependence

Definition: A set of vectors ({\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k}) is linearly dependent if there exist scalars (c_1, \ldots, c_k), not all zero, such that:
[c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k = \mathbf{0}]

The set is linearly independent if the only solution to this equation is (c_1 = c_2 = \cdots = c_k = 0).

Intuition: Linear dependence means at least one vector can be written as a linear combination of the others. Linear independence means no vector is redundant.

Example (dependent): In (\mathbb{R}^2), the vectors ((1,2)), ((2,4)) are dependent because (2(1,2) + (-1)(2,4) = (0,0)).

Example (independent): The standard basis vectors ((1,0)), ((0,1)) in (\mathbb{R}^2) are independent: the only way (c_1(1,0) + c_2(0,1) = (0,0)) is if (c_1 = c_2 = 0).

Key fact: Any set containing the zero vector is linearly dependent. Any set with more vectors than the dimension of the space is linearly dependent.