information:classification, stylistic:style_transfer
Bases, vector coordinates, and subspaces

Bases are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called Hamel bases, depends on the axiom of choice. It follows that, in general, no base can be explicitly described. For example, the real numbers form an infinite-dimensional vector space over the rational numbers, for which no specific basis is known.

Consider a basis (b 1, b 2, …, b n) {\displaystyle (\mathbf {b} _{1},\mathbf {b} _{2},\ldots,\mathbf {b} _{n})} of a vector space V of dimension n over a field F. The definition of a basis implies that every v ∈ V {\displaystyle \mathbf {v} \in V} may be written v = a 1 b 1 + ⋯ + a n b n, {\displaystyle \mathbf {v} =a_{1}\mathbf {b} _{1}+\cdots +a_{n}\mathbf {b} _{n},} with a 1, …, a n {\displaystyle a_{1},\dots,a_{n}} in F, and that this decomposition is unique. The scalars a 1, …, a n {\displaystyle a_{1},\ldots,a_{n}} are called the coordinates of v on the basis. They are also said to be the coefficients of the decomposition of v on the basis. One also says that the n - tuple of the coordinates is the coordinate vector of v on the basis, since the set F n {\displaystyle F^{n}} of the n -tuples of elements of F is a vector space for componentwise addition and scalar multiplication, whose dimension is n.

The one-to-one correspondence between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a vector space isomorphism, which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates.