Definition: A set V is a vector space over a field F, if given an operation vector addition defined in V, denoted v+w for all v, w in V, and an operation scalar multiplication in V, denoted a*v for all v in V and a in F, the following 10 properties hold for all a, b in F and u, v, and w in V:
- v+w belongs to V.
V is closed under vector addition.
- u+(v+w)= (u+v)+w.
Associativity of vector addition in V.
- There exists an element 0 in V, such that for all elements v in V, v+0=v.
Existence of an additive identity element in V.
- For all v in V, there exists an element -v in V, such that v+(-v)=0.
Existence of additive inverses in V.
Commutativity of vector addition in V.
- a*v belongs to V.
V is closed under scalar multiplication.
Associativity of scalar multiplication in V.
- If 1 denotes the multiplicative identity of the field F, then 1*v=v.
Neutrality of one.
Distributivity with respect to vector addition.
Distributivity with respect to field addition.
Properties 1 through 5 indicate that V is an Abelian group under vector addition. Properties 6 through 10 apply to scalar multiplication of a vector v in V by a scalar a in F. (Note that Property 5 actually follows from the other 9.)
From the above properties, one can immediately prove the following handy formulas:
- a*0 = 0*v = 0
- -(a*v) = (-a)*v = a*(-v)
for all a in F and v in V.
The members of a vector space are called vectors. The concept of a vector space is entirely abstract like the concepts of a group, ring, and field. To determine if a set V is a vector space one must specify the set V, a field F and define vector addition and scalar multiplication in V. Then if V satisfies the above 10 properties it is a vector space over the field F.
- /Example I: The vector space Rn, over R, with component-wise operations
- More generally, Fn, over F, with component-wise operations
- /Example II: The set of (mxn) matrices with complex elements over C
- More generally, the set of (mxn) matrices over an arbitrary field F
- /Example III: The set of all continuous real-valued functions on a closed interval
- Given a vector space V over F, and some set X, then the set of all functions X -> V forms a vector space over F
- The finite field GF(pn), over GF(p)
- C, over R
- R, over Q (the rational numbers)
Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set of vectors is called their span; if no vector can be removed without diminishing the span, the set is called linearly independent. A linearly independent set whose span is the whole space is called a basis.
All bases for a given vector space have the same cardinality. Using Zorn's Lemma, it can be proved that every vector space has a basis, and vector spaces over a given field are fixed up to isomorphism by a single cardinal number (called the dimension of the vector space) representing the size of the basis. For instance the real vector spaces are just R0, R1, R2, R3, ..., R∞, ... As you would expect, the dimension of the real vector space R3 is three.
A morphism from a vector space V to a vector space W (necessarily over the same field) is called a linear transformation or "linear map". That is, a map is linear if and only if it preserves sums and scalar products. An isomorphism is a linear map that is one-to-one and onto. The set of all linear maps from V to W is denoted L(V,W) and makes up a vector space over the same field. When bases for both V and W are given, linear maps can be expressed in terms of components as matrices.