Vector Space - Bases and Dimension

Bases and Dimension

Bases allow the introduction of coordinates as a means to represent vectors. A basis is a (finite or infinite) set B = {v_i}_{i ∈ I} of vectors v_i, for convenience often indexed by some index set I, that spans the whole space and is linearly independent. "Spanning the whole space" means that any vector v can be expressed as a finite sum (called a linear combination) of the basis elements:

(1)

where the a_k are scalars, called the coordinates of the vector v with respect to the basis B, and v_ik (k = 1, ..., n) elements of B. Linear independence means that the coordinates a_k are uniquely determined for any vector in the vector space.

For example, the coordinate vectors e₁ = (1, 0, ..., 0), e₂ = (0, 1, 0, ..., 0), to e_n = (0, 0, ..., 0, 1), form a basis of Fn, called the standard basis, since any vector (x₁, x₂, ..., x_n) can be uniquely expressed as a linear combination of these vectors:

(x₁, x₂, ..., x_n) = x₁(1, 0, ..., 0) + x₂(0, 1, 0, ..., 0) + ... + x_n(0, ..., 0, 1) = x₁e₁ + x₂e₂ + ... + x_ne_n.

The corresponding coordinates x₁, x₂, ..., x_n are just the Cartesian coordinates of the vector.

Every vector space has a basis. This follows from Zorn's lemma, an equivalent formulation of the Axiom of Choice. Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the axiom of choice. The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or cardinality (cf. Dimension theorem for vector spaces). It is called the dimension of the vector space, denoted dim V. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.

The dimension of the coordinate space Fn is n, by the basis exhibited above. The dimension of the polynomial ring F introduced above is countably infinite, a basis is given by 1, x, x2, ... A fortiori, the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite. Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of a homogeneous ordinary differential equation equals the degree of the equation. For example, the solution space for the above equation is generated by e−x and xe−x. These two functions are linearly independent over R, so the dimension of this space is two, as is the degree of the equation.

A field extension over the rationals Q can be thought of as a vector space over Q (by defining vector addition as field addition, defining scalar multiplication as field multiplication by elements of Q, and otherwise ignoring the field multiplication). The dimension (or degree) of the field extension Q(α) over Q depends on α. If α satisfies some polynomial equation

q_nαn + q_n−1αn−1 + ... + q₀ = 0, with rational coefficients q_n, ..., q₀.

("α is algebraic"), the dimension is finite. More precisely, it equals the degree of the minimal polynomial having α as a root. For example, the complex numbers C are a two-dimensional real vector space, generated by 1 and the imaginary unit i. The latter satisfies i2 + 1 = 0, an equation of degree two. Thus, C is a two-dimensional R-vector space (and, as any field, one-dimensional as a vector space over itself, C). If α is not algebraic, the dimension of Q(α) over Q is infinite. For instance, for α = π there is no such equation, in other words π is transcendental.