Vector Space Axioms a Real Vector Space
Vector Space Axioms 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 .
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