A real vector space (that is, a vector space over the real numbers R) is a set V of objects, individually called vectors, and two operations + (plus) and · (scalar multiplication), together satisfying the following axioms.
1. For every pair of vectors u, v ∈ V , the operation + is defined and gives a vector in V . That is, u + v ∈ V for all u, v ∈ V .
2. + is commutative, i.e., u + v = v + u for all u, v ∈ V .
3. + is associative, i.e., u +(v+w)=(u+v) + w for all u, v, w ∈ V .
4. + has at least one identity within V . A vector e is called an identity if, for all u ∈ V , we have u + e = u. In fact, our first theorem from these axioms will be that each vector space has exactly one identity. Therefore, it can be given a unique name, and the usual name is 0. (Use 0¯ when handwriting.)
5. Each v ∈ V has at least one additive inverse within V . That is, for each v ∈ V , there is at least one vector wv ∈ V such that v + wv is an identity. In fact, it is a theorem that every vector in every vector space has exactly one inverse, and so it can be given a special name. The usual name is −v. Think of this as an indivisible symbol; it doesn’t mean minus times v.
6. Scalar multiplication is defined between every real number and every vector in V , and the result is in V . That is, kv ∈ V for all k ∈ R and v ∈ V .
7. a(u + v) = au + av (scalar multiplication distributes over vector addition)
8. (a+b)v = av + bv (scalar multiplication distributes over real addition)
9. 1v = v (the real number 1 is a scalar multiplication identity)
10. (ab)v = a(bv)
Vectors are usually denoted in print by lowercase bold letters, as above; in handwriting one typically puts a bar or arrow over the letter, like this:u, ¯ ~u. The set of all vectors is usually denoted by a capital roman letter, but sometimes script capitals are used (V). The elements of R are called scalars and are usually denoted by lowercase roman letters, but sometimes lowercase Greek letters. Note also that the dot for scalar multiplication is usually omitted. In order to prove something for all vector spaces, one must prove it from the axioms, or from theorems previously proved from the axioms. You must be careful not to use other algebraic properties that seem obvious, unless you have already proved them from the axioms. Nor can you use any assumptions about what the vectors “look like”. For instance, a very important n vector space we will use a lot is R , the set of all real n-tuples (a1, a2,...,an). But if you assume in a proof that the vectors are n-tuples, at best you have proved your theorem for Rn,
Math 28S, S Maurer, October 6, 2010 Over −→ VectorSpaceAxioms Math28S,SMaurer, October6,2010 not all vector spaces. Note: When talking about Rn, n is arbitrary but fixed. So, if we prove something about Rn we have proved it for a lot of vector spaces, namely R1,R2,R3,.... But there are many more vector spaces than that. In proving theorems about vector spaces, there are a few things you are allowed to use in addition to the axioms above. You are allowed to use the rules of logic, of course. (E.g., if A implies B and B implies C, then A implies C.) You may also use the rules of equality. For instance, if A = B and B = C then A = C. Also, if A = B, then B may be substituted for A in any true statement involving A. Also, if you do the same thing to both sides of an equality (e.g., add u on the right to both sides, or multiply both sides by scalar k), the results are equal. You are allowed to use such steps because they are logic properties and thus may be used throughout mathematics. Axioms 1–5 say that V is a commutative group under +. Groups, commutative or not, are the first thing you study in the course Modern Algebra. Axioms 7–10 don’t have “for all” in them, and officially they should. E.g., Axiom 7 should include for all a ∈ R and u, v ∈ V . But in fact, the convention in mathematical writing is that when an equation is stated using variables, “for all” is understood unless the contrary is stated. In other words, the use of “for all” in Axioms 1–6 could have been omitted. Axioms 1 and 6 specify the nature of + and · as functions. There is a useful function notation that allows a much more succinct statement of these axioms, namely, the notation f : A → B. This notations says that f is a function with domain A and codomain B. (I.e., inputs are from A, outputs are in B. Older words for codomain are range and image space.) For instance, in school you could have said
Consider the function f : R → R defined by f(x) = x2.
This tells you that you are allowing any real number to be squared, and noting that the result is also always a real number. We need one more piece of notation to describe + and · as functions. The cartesian product A×B is the set of ordered pairs (a, b), where a ∈ A, b ∈ B. We now rewrite 1. and 6. as follows:
1′. +: V ×V → V
6′. · : R×V → V
Thus Axiom 1 says that the domain of the plus function is all pairs of vectors from V (we say + isa binary operation on V ), and that the codomain is also V . We say V is closed under +, because you never leave V when you take sums. From Axiom 6 we also say that V is closed under scalar multiplication, because multiplying some v ∈ V by any scalar k ∈ R never takes you out of V . The reals R are a “field”. There are lots of other fields, e.g., the complex numbers. One can also have a vector space over other fields. We may define “field”, and return to this generalization, later.
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