When using the standard basis, how is the representation vector described?

In the context of linear algebra, when using the standard basis for a vector space, the representation vector refers to how a vector can be expressed in terms of the basis. The standard basis consists of unit vectors aligned with the axes of the space being considered. For example, in (\mathbb{R}^n), the standard basis vectors are typically (e_1 = (1, 0, 0, \ldots, 0)), (e_2 = (0, 1, 0, \ldots, 0)), and so on up to (e_n).

When a vector has a representation in terms of the standard basis, it is described by its components corresponding to each of the basis vectors. This means that the representation vector is directly equal to the vector itself in the context of the coordinate system defined by the standard basis. For instance, if you have a vector (v = (v_1, v_2, \ldots, v_n)), in standard basis, it is represented as (v_1 e_1 + v_2 e_2 + \ldots + v_n e_n). Therefore, in this framework, the representation vector can