coordinate space. The case is the above-mentioned simplest example, in which the field is also regarded as a vector space over itself. The case and (so R2) was discussed in the introduction above. Complex numbers and other field extensions The set of complex numbers , that is, numbers that can be written in the form for real numbers and where is the imaginary unit, form a vector space over the reals with the usual addition and multiplication: and for real numbers , , , and . The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. In fact, the example of complex numbers is essentially the same as (that is, it is isomorphic to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number as representing the ordered pair in the complex plane then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example. More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: a field containing a smaller field is an -vector space, by the given multiplication and addition operations of . For example, the complex numbers are a vector space over , and the field extension is a vector space over . Function spaces Functions from any fixed set to a field also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions and is the function given by and similarly for multiplication. Such function spaces occur in many geometric situations, when is the real line or an interval, or other subsets of . Many notions in topology and analysis, such as continuity, integrability or differentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property. Therefore, the set of such functions are vector spaces, whose study belongs to functional analysis. Linear equations Systems of homogeneous linear equations are closely tied to vector spaces. For example, the solutions of are given by triples with arbitrary and They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. Matrices can be used to condense multiple linear equations as above into one vector