As the title implies, this short chapter serves as a reference. The nature of vectors has been extracted from discussions in previous chapters of this volume D of geometric position vectors and algebraic array vectors with their norms and dot products. However there are other vector spaces, norm, inner products. The symbols u,v,w will be used in the chapter to denote more general vectors, which may or may not be different from geometric position vectors and algebraic arrays p,q,r.
[1.1] (Basic axioms for any vector space) A vector space over the real numbers is a set V of elements called vectors which have the following properties (for any vectors u,v,w):
(a) There is an operation +, called vector addition such that the sum u + v is a vector in V.
(b) Vector addition is commutative: u + v = v + u.
(c) Vector addition is associative: (u + v) + w = u + (v + w).
(d) There is a zero vector 0 that is the additive identity: v + 0 = v.
(e) There is an additive inverse − v (negative) such that v + (− v) = 0.
Also (for any real numbers λ, σ):
(f) There is an operation called multiplication by scalars: the products λv , vλ are vectors in V
(g) Multiplication by scalars is commutative: λv = vλ
(h) Multiplication by scalars is associative: (λσ)v = λ(σv)
(i) The number 1 acts like an identity element with multiplication with vectors: 1v = v
(j) There is a left distributive law: λ(u + v) = λu + λv
(k) There is a right distributive law: (λ + σ)v
= λv + σv
Example1: Arrays of real numbers
All arrays (discussed in Chapater 4) of the same size form a vector space.
Example2: polynomials
Some integral powers of x are x0, x1, x2, x3, x4, ..... . In most of these discussions x0 and x1 are replaced by 1 and x respectively. A term is a product of a real number (called coefficient) and some power of x. A polynomial is a finite sum of terms (usually written with terms with decreasing powers of x). Terms with zero coefficients may be omitted from writings. For example,
Example3: Infinite Sequences
Infinite sequences are infinite but ordered collections of real numbers (therefore they are almost identical to arrays of infinite size). For example,
Example4: Functions from a non-empty set into the real numbers
Let S be any set except the empty set. Then there are many functions from the entire set S into the set of real numbers R. If f and g are such functions, then they carry any element x in S into R. The images are f(x) and g(x). The functions can be added by adding their images: the image of f + g is f(x) + g(x). This can be represented by the symbol (f + g)(x) = f(x) + g(x). For example,
αu + βv + γw is an example of a linear combination of vectors u,v,w. The zero vector is still denoted by 0.
The following definitions are not part of the basic definition of a vector space, but are additions to it. Most useful vector spaces have norms [3.1] and inner products [4.1] defined on them. The definitions of these extras including vector products are listed here for reference, but are discussed more fully in chapter 4 of this volume on vectors.
[2.1] Axioms for a norm on a vector space:
(a) the norm of any vector is a non-negative real number:
|v| ≥ 0
(b) Only the zero vector has norm zero:
|v| = 0 if and only if v = 0
(c) the norm of a product of a real number and a vector is equal to the product of the absolute value of the number and the norm of the vector:
|λv| = |λ| |v|
(d) (triangle inequality) the norm of a sum of two vectors cannot exceed the sum of their individual norms:
|u + v| ≤ |u| + |v|
A vector space with a norm is called a normed vector space.
Example1: Norms of geometric position vectors and algebraic arrays
These norms were discussed in Chapter 4 of this volumn (D).
Example2 Strange norm on arrays of size 2
If u = (x,y), then |u| = |(x,y)| = |x| + |y|
Click here to see the proof that this strange norm satisfies all for conditions for a norm.
[2.2] (Parallelogram law) A norm on a vector space is said to satisfy the parallelogram law if it satisfies the equation
It is called "the parallelogram law" because there is an equivalent equation in plane geometry about parallelograms. (Click here to see the equation and its proof.) The norms of lengths of position vectors and norms of arrays discussed in Chapter 4 satisfy the equation. Even the strange norm defined in Example2 above satisfies the parallelogram law. But there exist "weird" vector spaces with norms that do not satisfy the parallelogram law.
[4.1] Axioms for an inner product (u⋅v)
(a) The inner product is a real number (not a vector): u⋅v is a real number
(b) The inner product of a vector with itself is a non-negative real number: u⋅v ≥ 0
(c) The inner product of a vector with itself is zero if and only if the vector is the zero vector: v⋅v = 0 if and only if v = 0
(d) The inner product is commutative: u⋅v = v⋅u
(e) The inner product is distributive: u⋅(v + w) = u⋅v + u⋅w
(f) A real may be factored out of the geometric product: (λu)⋅v = λ(u⋅v)
A vector space with an inner product is called an inner product space.
The dot product discussed in the previous chapter (Chapter 4) of this volume (D) satisfied a list there ([2.2]) of properties almost identical to [4.1] given above. Therefore, the dot product is an inner product on arrays. An outer product (or vector product) is discussed in the next chapter (6). It is unique because of its limitation to arrays of size 3, and does not have the general nature of concepts discussed here.
[4.2] (Norm from inner product) For any vector space with an inner product, the positive square root of the inner product of any vector with itself has all the properties of a norm. That norm is said to be generated by the inner product.
Notation: |v| = +sqrt(v⋅v) is a norm.
Click here to see the argument supporting [4.2].
[4.3] (Generated norm and the parallelogram law) A norm generated by any inner product must satisfy the following equation:
for any vectors u,v, |u + v|2 + |u − v|2 = 2|u|2 + 2|v|2 [parallelogram law]
[4.4] (Schwarz inequality) The absolute value of an inner product of two vectors cannot be larger than the product of the norms of the vectors.
Notation: for any vectors u,v, |u⋅v| ≤ |u| |v|.
Click here to see a proof of [4.4] for any inner product space, i.e. without using geometry or arrays.
The Schwarz inequality makes possible part (b) of the following:
[4.5] Abstract geometric properties defined for vector spaces with an inner product:
(a) u and v are perpendicular if and only if u⋅v = 0
(b) the cosine of the angle between non-zero vectors u and v = u⋅v/(|u| |v|)