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 e_{1} = (1, 0, ..., 0), e_{2} = (0, 1, 0, ..., 0), to e_{n} = (0, 0, ..., 0, 1), form a basis of Fn, called the standard basis, since any vector (x_{1}, x_{2}, ..., x_{n}) can be uniquely expressed as a linear combination of these vectors:
 (x_{1}, x_{2}, ..., x_{n}) = x_{1}(1, 0, ..., 0) + x_{2}(0, 1, 0, ..., 0) + ... + x_{n}(0, ..., 0, 1) = x_{1}e_{1} + x_{2}e_{2} + ... + x_{n}e_{n}.
The corresponding coordinates x_{1}, x_{2}, ..., 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_{n}αn + q_{n−1}αn−1 + ... + q_{0} = 0, with rational coefficients q_{n}, ..., q_{0}.
("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 twodimensional 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 twodimensional Rvector space (and, as any field, onedimensional 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.
Read more about this topic: Vector Space
Famous quotes containing the words bases and/or dimension:
“The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)
“By intervening in the Vietnamese struggle the United States was attempting to fit its global strategies into a world of hillocks and hamlets, to reduce its majestic concerns for the containment of communism and the security of the Free World to a dimension where governments rose and fell as a result of arguments between two colonels’ wives.”
—Frances Fitzgerald (b. 1940)