basis

A set \mathcal B=\{\mathbf b_1,\dots,\mathbf b_k\} in a vector space V is a basis for V if it spans V and is linearly independent. These two conditions together force a third: every \mathbf v\in V has a unique representation as a combination of \mathcal B's vectors. A basis is not unique to a space, but once one is fixed, coordinates relative to it are locked in.