Independence, Basis, and Dimension

Vectors $\mathbf{v}_1, \dots, \mathbf{v}_k$ are linearly independent if the only combination giving zero is the trivial one: $c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k = \mathbf{0} \implies c_i = 0$ for all $i$. Equivalently: no vector in the list is a combination of the others, so nothing is redundant.

Three vectors in $\mathbb{R}^2$ are always dependent: the plane doesn't have room for three independent directions. In general, more than $n$ vectors in $\mathbb{R}^n$ must be dependent.