Read an Excerpt

Elements Of Tensor Calculus

Dover Publications, Inc.

Vector Spaces

I. CONCEPT OF A VECTOR SPACE

1. Definition of a vector space. Consider the set of displacement vectors of elementary vector analysis. These satisfy the following rules:

(i) The result of vector addition of any two vectors, x and y, is their vector sum, or resultant, x + y. Vector addition has the following properties:

(a) x + y = y + x (commutative property);

(b) x + (y + z) = (x + y) + z (associative property);

(c) there exists a zero vector denoted by ) 0such that x + 0 = x;

(d) for every vector ) x there is a corresponding negative vector (–x), such that x + (–x) = 0.

(ii) The result of multiplying a vector x by a real scalar α is a vector denoted by αx. Scalar multiplication has the following properties:

(a') x = x;

(b') α(βx) = (αβ)x (associative property);

(c') (α + β)x = αx + βx (distributive property for scalar addition);

(d') α(x + y) = αx + αy (distributive property for vector addition).

Using the above properties as a guide, we now consider a general set E of arbitrary elements x, y etc., which obey the following rules:

(1) To every pairx, y, there corresponds an element x + y having the properties (a), (b), (c), (d).

(2) To every combination of an elementxand a real number a there corresponds an element αxhaving the properties (a'), (b'), (c'), (d').

We then say that E is a vector space over the field of real numbers and that the elementsx, y, etc., are vectors in E. If the second rule holds for all complex numbers a then E is a vector space over the field of complex numbers. Except when otherwise stated we shall confine ourselves in this book to the study of vector spaces over the field of real numbers.

2. Examples of vector spaces. There are several other simple examples of vector spaces which may be quoted to give an idea of the interest and application of the general concept.

(a) Consider the set of complex numbers a + ib, where a and b are real. The addition of any two complex numbers (a + ib, c + id, etc.) and the multiplication of a complex number by a real number a obviously have the properties listed in §1. It flows that the set of complex numbers constitutes a vector space over the field of real numbers.

(b) Let X be an array of n real numbers arranged in definite order

X = (x1, x2, ..., xn)

and let E be the set of all arrays X. Assume the following two rules of composition:

If X = (x1, x2, ..., xn) and Y = (y1, y2, ...,yn) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If X = (x1, x2, ..., xn) and if a is any real number then

αX = (αx1, αx2, ..., αxn).

It is easily verified that these two rules imply the rules (1) and (2) of §1. It then follows that E constitutes a vector space with respect to the field of real numbers.

(c) Consider the set of real functions of a real variable defined on the interval (0, 1) with the usual composition rules for the sum of two functions and for the product of a function by a constant α. With these rules the set under consideration is a vector space over the field of real numbers.

3. Elementary properties of vector spaces. (1) For any two vectors x and y there is one, and only one, vector z such that

x = z + y (3.1)

This is easily seen by adding the vector (-y) to each side of (3.1) giving the relation

z = x + (-y)

which defines z uniquely. As in elementary algebra we write

x + (-y) = x - y.

With this notation the property (c') of §1 can be written as

(α - β)x = αx - βx. (3.2)

In view of this property it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Putting α = β it can immediately be deduced from (3.2) that

0x = 0 (3.3)

and, on writing α = 0

(-β)x = -βx.

In particular

(-1)x = -x (3.4)

(2) From (3.4) it follows that the property (d') of §1 can be rewritten in the form

α(x-y) = αx - αy. (3.5)

Putting x = y in (3.5) it follows that

α0 = 0 (3.6)

(3) Conversely, the relation

αx = 0 (3.7)

implies that either α = 0 or x = 0. For, if a is not zero it has an inverse a1, and on multiplying both sides of (3.7) by a-1 we have

α-1 (αx) = 0,

or

(α-1 α)x = x = 0,

which is the required result.

4. Vector sub-spaces.Definition: A sub-space of a vector space E is any part, V, of E which is such that, if x and y belong to V and a is any real number, then the vectors x + y and ax also belong to V.

The commutative, associative and distributive properties of E clearly apply to V. The real number α may be zero so that it is clear that V necessarily contains the zero vector. Again if x belongs to V, (-l)x = -x also belongs to V and it follows that each vector of V has a negative in V. The rules of addition and of multiplication by a scalar thus have the properties listed in §1, therefore V itself is a vector space.

Some simple examples of vector sub-spaces may be given.

(a) The set of vectors coplanar with two given vectors constitutes a sub-space of the vector space of elementary geometry.

(b) If x is a non-zero vector of a vector space E, the set of products αx, where αis any real number, constitutes a subspace of E.

(c) The set of real functions of a real variable defined on the interval (0, 1) forms a vector space over the field of real numbers. The bounded functions of a real variable defined in the same way form a sub-space of this vector space since the sum of two bounded functions and the product of a bounded function by a constant are themselves bounded.

II. n-DIMENSIONAL VECTOR SPACES

5. Basis of a vector space. Let x1, x2, ..., xp be p non-zero vectors in a vector space E. These vectors are said to form a linearly independent system of order p if it is impossible to find p numbers α1, α2, ..., αp, not all zero, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the contrary case the given system of p vectors is said to be linearly dependent.

Consider the set of all systems of linearly independent vectors in the vector space E. There are two possibilities - either (a) there exist linearly independent systems of arbitrarily large order, or (b) the order of the linearly independent systems is bounded.

In the second case the vector space is said to have a finite number of dimensions. This classification will be explained shortly. In the remainder of this book we shall only consider vector spaces which have a finite number of dimensions. In this case it is possible to determine an integer n such that there exist linearly independent systems of order n but not of order (n + 1). If (e1, e2, ..., en) is any such system of order n it will be called a basis of E in conformity with the following definition:

The basis of a vector space E is any linearly independent system of maximum order.

Let x be any vector in E. The system of (n + 1) vectors (x, e1, e2, ..., en) is necessarily linearly dependent, so there exist (n + l) numbers λ, a1, a2, ..., an such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

λ must be different from zero, otherwise the system ei will not be linearly independent. Equation (5.1) can thus be solved for x and there exist n numbers x1, x2, ..., xn such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.2)

and the vector x is expressible as a linear combination of the ei. Moreover, this combination is unique for, if there existed another, their difference would constitute a linear combination of the ei equal to the null vector and with coefficients not all zero. This conflicts with the original postulates. This result may be stated formally:

THEOREM: Given a basis of E, any vectorxof E can be represented in a unique way as a linear combination of the vectors of this basis.

The numbers (x1, x2, ..., xn) which appear in (5.2) are called the components of x with respect to the basis (e1, e2, ..., en).

It is easy to show that the property stated in the above theorem specifies bases amongst all systems of vectors. Let (e1, e2, ..., ep) be a system of p vectors such that any vector x ofE can be expressed uniquely in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

In particular the null vector 0 can be expressed in one way only (x1 = x2 = ... = xp = 0) as a linear combination of the vectors of the system. It follows that such a system must be linearly independent and that p ≤ n.

It is clear that the same property holds for all linearly independent systems of order p. Let ([member of]1, [member of]2, ..., [member of]p) be such a system. [member of]1 can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Suppose that a1 (for example) is non zero, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and on substituting in (5.3) it is seen that any vector x of E can be expressed as a linear combination of ([member of]1, e2, ..., ep). In particular

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with at least one of the numbers y2, ..., yp pdifferent from zero, otherwise the system of [member of]i's cannot be linearly independent. Repeating this procedure we find that any vector x of E can be expressed in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows immediately that there cannot exist any linearly independent systems of order (p + l), consequently p = n. We now express this formally:

THEOREM: For a system of vectors to constitute a basis of E it is necessary and sufficient that any vector of E can be expressed in one, and only one, way as a linear combination of the vectors of that system.

The number n is termed the dimension of the vector space under consideration. We shall, in future, use En to denote an n-dimensional vector space.

6. Examples. (a) In the vector space of elementary geometry a basis is formed by any three non-coplanar vectors. This space is therefore three dimensional.

(b) Take the example (b) of §2 and consider the vectors

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.1)

Any vector

x = (x1,x2, ..., xn)

can be expressed in one, and only one, way as a linear combination of the ei:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that these vectors constitute a basis for E, which is thus an n-dimensional space.

7. Vector sub-spaces ofEn. Let V be a vector sub-space of En. Then V is a vector space in which any linearly independent system of vectors is also a linearly independent system of En. It follows that V has a finite number of dimensions r [??] n n. Let ([member of]1, [member of]2, ..., [member of]r) be a basis of V. Any vector of V can be put in one, and only one, way into the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.1)

Whatever the numbers [xi]1, [xi]2, ..., [xi]r, a vector in the form of (7.1) belongs to V.

Conversely, if ([member of]1, [member of]2, ..., [member of]r) is a linearly independent system of order r in En it is clear that the set of all vectors which can be expressed as a linear combination of the [member of]i constitute a vector sub-space of En.

Any n-dimensional vector sub-space of En coincides with En.

8. Complementary vector sub-spaces. Let us consider a linearly independent system ([member of]1, [member of]2, ..., [member of]r) of order r< n. We wish to complete this system by adding (n - r) new vectors (ηr + 1, ..., ηn) such that the system of vectors [member of]i and ηj constitutes a basis of En.

There certainly exists in En at least one vector such that the system formed by adjoining it to the [member of]i is a linearly independent system; for if not, the system [member of]i would itself be a basis of En. Let ηr + 1 be this vector so that ([member of]1, [member of]2, ..., [member of]r, ηr + 1) is a linearly independent system of order (r + 1). Repeating this procedure gives systems of increasing order and it will stop only when the order reachesn, the dimensionality of En. We now express this formally:

THEOREM: Given a linearly independent system of order r it is always possible to complete the system with (n - r) vectors to obtain a basis for En.

Let Ur be a vector sub-space of En having r< n dimensions and let ([member of]l, [member of]2, ..., [member of]r) be any basis of Ur. Using the preceding theorem n vectors (ηr + 1, ..., ηn) can be found (in point of fact in an infinite number of ways) such that the [member of]i and ηj define a basis of En. Let the vector sub-space generated by the ηj be denoted by Vn - r.

Evidently the two sub-spaces Ur and Vn -r have only the zero vector in common. Moreover any vector x of En which is expressible in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

can be decomposed into the sum

x = y + z

of a vector y of Ur and a vector z of Vn - r. This decomposition is evidently unique according to the preceding remarks. In view of this property the vector sub-spaces Ur and Vn - r are said to be complementary. This may be expressed formally:

THEOREM: TO every vector sub-space Ur of En there corresponds a unique complementary sub-space Vn - r.

9. Change of basis. It follows from the first theorem of §8 that a vector space En has an infinity of bases. We propose to determine the relations that exist between the components of a particular vector x with respect to two distinct bases.

---

Excerpted from Elements Of Tensor Calculus by A. Lichnerowicz, J. W. Leech, D. J. Newman. Copyright © 2016 A. Lichnerowicz. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

---