Bases

Definition: A basis of a vector space (V) is a set of vectors ({\mathbf{v}_1, \ldots, \mathbf{v}_k}) that is:

1. Linearly independent, and
2. Spans (V) (i.e., (\text{span}{\mathbf{v}_1, \ldots, \mathbf{v}_k} = V))

Equivalently, a basis is a minimal spanning set or a maximal linearly independent set.

Standard basis of (\mathbb{R}^n):
[\mathbf{e}_1 = (1,0,\ldots,0), \quad \mathbf{e}_2 = (0,1,\ldots,0), \quad \ldots, \quad \mathbf{e}_n = (0,0,\ldots,1)]

Example: In (\mathbb{R}^2), the set ({(1,0), (0,1)}) is the standard basis. But ({(1,1), (1,-1)}) is also a basis.

Key theorem: All bases of a given vector space have the same number of elements. This number is called the dimension of the space.

Coordinates: Once a basis is chosen, every vector (\mathbf{v} \in V) can be written uniquely as (\mathbf{v} = c_1\mathbf{v}_1 + \cdots + c_k\mathbf{v}_k). The coefficients ((c_1, \ldots, c_k)) are the coordinates of (\mathbf{v}) with respect to that basis.