1 The Vector Space Axioms
A vector space over a field K is a set V equipped with two operations — addition +:V×V→V and scalar multiplication ⋅:K×V→V— satisfying the following eight axioms:
u+v=v+u (commutativity of addition).
(u+v)+w=u+(v+w) (associativity of addition).
There exists an element 0∈V such that v+0=v for all v∈V (existence of a zero vector).
For every v∈V, there exists −v∈V such that v+(−v)=0 (existence of additive inverses).
a(bv)=(ab)v (associativity of scalar multiplication).
1⋅v=v (identity element of scalar multiplication).
a(u+v)=au+av (distributivity over vector addition).
(a+b)v=av+bv (distributivity over scalar addition).
Here a,b∈K and u,v,w∈V.
Throughout this chapter, we work over the field K=R unless stated otherwise.
2 Fundamental Examples
Rn={(a1,…,an)∣ai∈R} is a vector space over R under componentwise addition and scalar multiplication. The zero vector is 0=(0,…,0).
K[x]≤n={a0+a1x+⋯+anxn∣ai∈K} is a vector space over K under polynomial addition and scalar multiplication. The zero vector is the zero polynomial 0, and dimK[x]≤n=n+1.
The set Mm×n(K) of all m×n matrices over K is a vector space under matrix addition and scalar multiplication. The zero vector is the zero matrix O, and dimMm×n(K)=mn.
The set C[a,b] of all real-valued continuous functions on the interval [a,b] forms a vector space over R under pointwise addition (f+g)(x)=f(x)+g(x) and scalar multiplication (cf)(x)=c⋅f(x). This is an infinite-dimensional vector space.
3 Uniqueness of the Zero Vector and Additive Inverses
The zero vector of a vector space V is unique.
Suppose 0 and 0′ are both zero vectors. Since 0 is a zero vector, 0′+0=0′. Since 0′ is a zero vector, 0′+0=0. Therefore 0′=0. □
For each v∈V, the additive inverse −v is unique.
Suppose w1 and w2 are both additive inverses of v. Then w1=w1+0=w1+(v+w2)=(w1+v)+w2=0+w2=w2. □
4 Basic Properties of Scalar Multiplication
In any vector space V, the following hold:
0⋅v=0 (the scalar 0 annihilates every vector).
a⋅0=0 (every scalar annihilates the zero vector).
(−1)v=−v.
av=0⟹a=0 or v=0.
(1) We have 0⋅v=(0+0)v=0v+0v. Subtracting 0v from both sides yields 0=0v.(2) Similarly, a⋅0=a(0+0)=a0+a0. Subtracting a0 from both sides gives 0=a0.(3) We compute v+(−1)v=1⋅v+(−1)v=(1+(−1))v=0v=0. By uniqueness of the additive inverse, (−1)v=−v.(4) Suppose a=0. Multiplying both sides of av=0 by a−1 gives v=a−10=0. □
5 Subspaces
A nonempty subset W of a vector space V is called a subspace of V if W is itself a vector space under the operations inherited from V.
Let V be a vector space and W⊆V a nonempty subset. Then W is a subspace of V if and only if the following two conditions hold:
u,v∈W⟹u+v∈W.
a∈K,v∈W⟹av∈W.
Necessity is immediate. For sufficiency, since W=∅, pick any v∈W. Setting a=0 in condition (2) gives 0=0⋅v∈W. Setting a=−1 gives −v∈W. The remaining axioms (associativity, commutativity, distributivity, etc.) are inherited from V. □
In R3, the set W={(x,y,z)∣x+y+z=0} is a subspace. Indeed, if u=(u1,u2,u3) and v=(v1,v2,v3) lie in W, then (u1+v1)+(u2+v2)+(u3+v3)=0, so u+v∈W. Closure under scalar multiplication is verified similarly.
The set {(x,y,z)∣x+y+z=1} is not a subspace, since the zero vector 0=(0,0,0) does not belong to it.
6 Sum Spaces and Direct Sums
Let W1 and W2 be subspaces of V. Their sum is defined as W1+W2={w1+w2∣w1∈W1,w2∈W2}.
This is the smallest subspace of V containing W1∪W2.
If every vector v∈W1+W2 can be written uniquely as v=w1+w2 with wi∈Wi, then the sum is called a direct sum and is denoted W1⊕W2.
The sum W1+W2 is a direct sum if and only if W1∩W2={0}.
(⇒) Let v∈W1∩W2. Then v=v+0=0+v provides two decompositions. By uniqueness, v=0.(⇐) Suppose v=w1+w2=w1′+w2′. Then w1−w1′=w2′−w2∈W1∩W2={0}, whence w1=w1′ and w2=w2′. □
For finite-dimensional subspaces W1,W2 of V, dim(W1+W2)=dimW1+dimW2−dim(W1∩W2).
In particular, if W1+W2 is a direct sum, then dim(W1⊕W2)=dimW1+dimW2.