Hassani S. Mathematical Methods: For Students of Physics and Related Fields

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214 Planar and Spatial Vectors

6.22. Acoordinatesystem(R, Θ,φ) in space is deﬁned by

x = R cos Θ cos φ + b cos φ

y = R cos Θ sin φ + b sin φ

z = R sin Θ

where b is a constant, and 0 <R<b. Using the Jacobian for three variables

(6.66), ﬁnd the element of volume for this coordinate system.

Chapter 7

Finite-Dimensional Vector

Spaces

Human visual perception of dimension is limited to two and three, the plane

and space. However, his mental perception, and his ability to abstract, rec-

ognizes no bounds. If this abstraction were a mere useless mental exercise,

we would not bother to add this chapter to the book. It is an intriguing

coincidence that Nature plays along with the tune of human mental abstrac-

tion in the most harmonious way. This harmony was revealed to Hermann

Minkowski in 1908 when he convinced physicists and mathematicians alike,

that the most natural setting for the newly discovered special theory of rel-

ativity was a four-dimensional space. Eight years later, Einstein used this

concept to formulate his general theory of relativity which is the only viable

theory of gravity for the large-scale structure of space and time. In 1921,

Kaluza, in a most beautiful idea, uniﬁed the electromagnetic interaction with

gravity using a ﬁve-dimensional spacetime. Today string theory, one of the

most promising candidates for the uniﬁcation of all forces of nature, uses

11-dimensional spacetime; and the language of quantum mechanics—a the-

ory that describes atomic, molecular, and solid-state physics, as well as all

of chemistry—is best spoken in an inﬁnite-dimensional space, called Hilbert

space.

The key to this multidimensional abstraction is Descartes’ ingenious idea

of translating Euclid’s geometry into the language of coordinates whereby the

abstract Euclidean point in a plane is given the two coordinates (x, y), and

that in space, the three coordinates (x, y, z), where x, y,andz are real num-

bers. Once this crucial step is taken, the generalization to multidimensional

spaces becomes a matter of adding more and more coordinates to the list:

(x, y, z, w) is a point in a four-dimensional space, and (x, y, z, w, u) describes

a point in a ﬁve-dimensional space. In the spirit of this chapter, we want to

identify points with vectors as in the plane and space, in which we drew a

216 Finite-Dimensional Vector Spaces

directed line segment from the origin to the point in question. In general, an

n-dimensional Cartesian vector x is

n-dimensional

Cartesian vector

x =(x

,...,x

) (7.1)

in which x

is called the jth component of the vector. These have all the

properties expected of vectors: You can add them

x + y =(x

,...,x

)+(y

,...,y

) ≡ (x

+ y

,...,x

+ y

you can multiply a vector by a number

αx = α(x

,...,x

) ≡ (αx

,αx

,...,αx

and the zero vector is 0 =(0, 0,...,0). Two vectors are equal if and only if

their corresponding components are equal. Sometimes, it will be convenient

to denote these vectors as columns rather than rows.

The set of real numbers, or the set of points on a line, is denoted by R.

It is common to denote the set of points in a plane—or, in the language of

Cartesian coordinates, the set of pairs of real numbers (x, y)—by R

,and

the set of points in space by R

. Generalizing this notation, we denote the

set of points in the n-dimensional Cartesian space by R

.Wenowhavean

inﬁnite collection of “spaces” of various dimensions, starting with the one-

dimensional real line R

= R, moving on to the two-dimensional plane R

and the three-dimensional space R

, and continuing to all the abstract spaces

with n ≥ 4. The concepts of linear combination, linear independence,andstandard basis of

basis are exactly the same as before. The vectors

≡ (1, 0,...,0),

≡ (0, 1,...,0), ...

≡ (0, 0,...,1) (7.2)

form a basis for R

, called the standard basis.

7.1 Linear Transformations

A linear transformation or a linear operator is a correspondence thatformal deﬁnition

of a linear

transformation or

a linear operator

takes a vector in one space and produces a vector in another space in such a

way that the operation of summation of vectors and multiplication of vectors

by numbers is preserved. If we denote the linear transformation by T,then

in mathematical symbolism, the above statement becomes

T(αx + βy)=αT(x)+βT(y). (7.3)

Matrices are prototypes of linear transformations. In fact, we saw earlier

that it was possible to transform vectors in the plane to vectors in space

and vice versa via 3 × 2or2× 3 matrices. We did not attempt to verify

Equation (7.3) for those transformations, but the reader can easily do so.

7.1 Linear Transformations 217

In fact, denoting vectors of R

and R

by column vectors, we can immediately

generalize Equations (6.36) and (6.37) to

⎛

⎜

⎝



⎞

⎟

⎠

⎛

⎜

⎝

... a

⎞

⎟

⎠

⎛

⎜

⎝

⎞

⎟

⎠

or a



= Aa, (7.4)

where A is an m × n matrix—i.e., it has m rows and n columns—whose

elements a

are real numbers. The reader may verify that Equation (7.4) is

a linear transformation that maps vectors of R

to those of R

Other linear operators of importance are various diﬀerential operators,

derivative is a

linear operator

i.e., derivatives of various order. For example, it is easily veriﬁed that d/dx

is a linear operator acting on the space of diﬀerentiable functions.

This is

because

(αf + βg)=α

+ β

for α and β real constants. Similarly d

/dx

and derivative of higher orders, as

well as partial derivatives of various kinds and orders, are all linear operators.

In fact, even when these derivatives are multiplied by functions (on the left),

they are still linear. In particular, the second-order linear diﬀerential operator

second-order linear

diﬀerential

operator

L ≡ p

(x)

+ p

(x)

+ p

(x)

is indeed a linear operator.

If a linear transformation T maps vectors of R

to vectors of R

,andS

maps vectors of R

to vectors of R

, then we can “compose” or “multiply”

the two transformations to obtain a linear transformation ST which maps

vectors of R

to vectors of R

. In terms of matrices, T is represented by an

m ×n matrix T, S is represented by a k ×m matrix S,andST is represented

by an k × n matrix which is the product of S and T with S to the left of T.

The product of matrices is as outlined in Box 6.1.3.

Box 7.1.1. If A is a k × m matrix, and B is an m × n matrix, then AB

is a k × n matrix whose entries are given by Box 6.1.3.

The product BA is not deﬁned unless k = n,inwhichcaseBA will be an

m × m matrix.

Using polynomials, we can generate multidimensional vector spaces by

polynomials can

generate

multidimensional

vector spaces!

adding increasing powers of t. Then, the collection P

[t] of polynomials of

degree n and less becomes an (n + 1)-dimensional vector space. A convenient

basis for this vector space is {1,t,t

,...,t

} which we call the standard

The reader may want to check that the collection of diﬀerentiable functions is indeed a

vector space with the “zero function” being the zero vector.

218 Finite-Dimensional Vector Spaces

basis of P

[t]. The reader may verify that the operation of diﬀerentiation (of

any order) is a linear transformation on P

[t] which can be represented by

matrices as done in Example 6.2.2.

Example 7.1.1.

Let us ﬁnd the matrix that represents the operation of second

diﬀerentiation on P

[t] using the standard basis of P

[t]. Recall that we only need to

apply the second derivative to the basis vectors f

=1,f

= t, f

= t

,andf

= t

We use a prime to denote the transformed vector:



(1) = 0 = 0 ·f

+0· f



(t)=0=0·f

+0· f



)=2=2·f

+0· f



)=6t =0· f

+6· f

where we have anticipated the fact that double diﬀerentiation of P

[t]resultsin

[t]. Following the rule of Box 6.2.2, we can write the transformation matrix as



0020

0006



We may verify that the coeﬃcients in P

[t] of the second derivative of an arbi-

trary polynomial f (t)=α

+ α

t + α

+ α

can be obtained by the product of

the matrix of second derivative and the 4 × 1 column vector representing f(t). In

fact,



0020

0006



⎛

⎜

⎝

⎞

⎟

⎠



2α

6α



These are the two coeﬃcients of the resulting polynomial in P

[t]. The polynomial

itself is 2α

+6α

t which is indeed the derivative of the third degree polynomial

f(t).



7.2 Inner Product

Since the concepts of length and angle are not familiar for R

, we need to

deﬁne the inner product ﬁrst and then deduce those concepts. We can gener-

alize the usual inner product of R

and R

intermsofcomponentsofvectors.

Let

a =(a

,...,a

)andb =(b

,...,b

Then

inner product in

deﬁned in

terms of

components in the

standard basis

a ·b = a

+ a

+ ···+ a

(7.5)

is the immediate generalization of the dot product to R

7.2 Inner Product 219

This, of course, is not the most general inner product. For that, we need

an inner product matrix G. As in the case of the plane and space, this is

simply a symmetric n ×n matrix whose elements determine the dot products

of the vectors of the basis in which we are working.

G =

⎛

⎜

⎝

... g

⎞

⎟

⎠

= g

,i,j=1, 2,...,n. (7.6)

Example 7.2.1.

Let us ﬁnd the inner product matrix for the basis {1,t,t

} of

[t]. As usual, we assume that the interval of integration for the inner product is

(0, 1). Because of the symmetry of the matrix and the fact that we have already

calculated the 3 × 3 submatrix of G, we need to ﬁnd g

, g

,andg

.Once

again, let f

= f

(t)=1,f

= f

(t)=t, f

= f

(t)=t

,andf

= f

(t)=t

;

then

= f

· f

(t)f

(t) dt =

dt =

= f

· f

(t)f

(t) dt =

dt =

Similarly, g

and g

. It follows that

G =

⎛

⎜

⎝

⎞

⎟

⎠

This matrix can be used to ﬁnd the dot product of any two vectors in terms of their

components in the basis {1,t,t

} of P

[t].



If a and b have components (a

,...,a

)and(b

,...,b

), then their

inner product is given by

inner product in

deﬁned in

terms of the

metric matrix and

components in a

general basis

aGb =

... a

⎛

⎜

⎝

... g

⎞

⎟

⎠

⎛

⎜

⎝

⎞

⎟

⎠

. (7.7)

As usual, if this expression is zero, we say that a and b are G-orthogonal. For

an orthonormal basis, the inner product matrix G becomes the unit matrix

and we recover the usual inner product of vectors in terms of components.

With a positive deﬁnite inner product at hand, we can deﬁne the length of

length of a vector

deﬁned in terms of

inner product

a vector as the (positive) square root of the inner product of the vector with

itself. Can we deﬁne the angle as well? We can always deﬁne

Only if the inner product is positive deﬁnite.

220 Finite-Dimensional Vector Spaces

angle deﬁned in

terms of inner

product

cos θ ≡

a ·b

|a||b|

a ·b

√

a ·a

√

b · b

But how do we know that the ratio on the RHS is less than one? After all,

a true cosine must have this property! It is an amazing fact of nature that

any positive deﬁnite inner product has precisely this property. To show this,

let a and b be two vectors in any vector space on which an inner product is

deﬁned. Denote the unit vector in the a direction by

, and construct the

vector



= b − (b ·

)



 

anumber

(7.8)

which is easily seen to be perpendicular to

(and therefore to a). If the

inner product is positive deﬁnite, then

derivation of the

Schwarz inequality



· b



≥ 0 ⇒ [b − (b ·

)

] · [b − (b ·

)

] ≥ 0

b ·b





=|b|

−2 b ·[(b ·

)

]



 

=(b·

)

+(b ·

)

  

≥ 0.

It follows that

|b|

− (b ·

)

≥ 0 ⇒|b|

≥



b ·



|a|



and

|b|

≥



b · a

|a|



⇒|b|

|a|

≥ (b ·a)

This is the desired inequality.

Box 7.2.1. (Schwarz Inequality).Ifa and b are two nonzero vectors

of a vector space for which a positive deﬁnite inner product is deﬁned,

then

|a||b|≥|a ·b|.

The equality holds only if b is a multiple of a.

The last statement follows from the fact that b



· b



= 0 only if b



=0when

the inner product is positive deﬁnite [see Equation (7.8)].

The Schwarz inequality holds not only for ﬁnite-dimensional vector spaces

Schwarz inequality

holds in all inner

product spaces

regardless of their

dimensionality.

such as R

or P

[t], but also for inﬁnite-dimensional vector spaces. It is

one of the most important inequalities in mathematical physics. One of its

consequences is that we can actually deﬁne the angle between two nonzero

vectors in R

or P

[t] (or any other vector space, ﬁnite or inﬁnite, for which

a positive deﬁnite inner product exists).

7.2 Inner Product 221

Example 7.2.2. What is the angle between the third and fourth vectors in the

standard basis of P

[t] when the interval of integration of the inner product is (0, 1)?

All the inner products are calculated in Example 7.2.1. Therefore,

cos θ =

· f

√

· f

√

· f

√

1/6



1/5



1/7

√

or θ =9.594

◦



As in the case of the plane and space, it is convenient to construct or-

thonormal basis vectors in R

. This is done by the Gram–Schmidt process Gram–Schmidt

process

which can easily be generalized. Suppose B = {a

, a

,...,a

} is a basis for

. Again, to avoid complications, we assume that the inner product is Eu-

clidean so that the inner product of every nonzero vector with itself is positive.

We know how to construct three orthonormal vectors out of {a

, a

},we

did that in our discussion of the space vectors. Call these new orthonormal

vectors {

}. Now construct the vector a



= a

− (a

)

− (a

)

− (a

)

which is obtained from a

by taking away its projections along

,and

Now note that

·a



· a

− (a

)

  

−(a

)

  

−(a

)

  

=0.

Similarly,

· a



=0and

· a



= 0; i.e., a



is orthogonal to

,and

This suggests deﬁning

≡







· a



This process can continue until we come up with n orthonormal vectors. This

will happen only if the n vectors with which we started are linearly indepen-

dent.

Box 7.2.2. If {a

, a

,...,a

} are linearly independent vectors of R

then we can construct a set of n orthonormal vectors out of them by the

Gram–Schmidt process.

An orthonormal basis will be denoted by {

,...,

},where,asusual,

the symbol

e stands for unit vectors. We can abbreviate the orthonormal

property of these vectors by writing

1ifi = j,

0ifi = j.

222 Finite-Dimensional Vector Spaces

There is a symbol that shortens the above statement even further. It is called

the Kronecker delta and denoted by δ

. It is deﬁned byKronecker delta

and its use in

discussing

orthonormal

vectors

1ifi = j,

0ifi = j.

(7.9)

Therefore, the orthonormality condition can be expressed as

= δ

. (7.10)

We shall see many examples of the use of the Kronecker delta in the sequel.

Transformations that leave the inner products unchanged can be obtained

in exactly the same way as for the plane and the space. For A to preserve the

inner product, we need to have

G-orthogonal

matrix

AGA = G, (7.11)

i.e., it has to be G-orthogonal. If G is the identity matrix, then A can be

thought of as an n-dimensional rigid rotation and is simply called orthogonal;

it satisﬁes

AA = 1 (7.12)

⎛

⎜

⎝

... a

⎞

⎟

⎠

⎛

⎜

⎝

... a

⎞

⎟

⎠

⎛

⎜

⎝

10... 0

01... 0

00... 1

⎞

⎟

⎠

It should be clear from this that the columns of the matrix A,consideredas

vectors, have unit length and are orthogonal to other columns in the usual

Euclidean inner product.

7.3 The Determinant

The determinant of an n×n matrix is obtained in terms of cofactors in exactly

the same way as in the case of 3 × 3 matrices. The cofactors are themselves

determinants of (n − 1) × (n − 1) matrices which can be expanded in terms

of cofactors of their elements which are determinants of (n − 2) × (n − 2)

matrices, etc. Continuing this process, we ﬁnally end up with determinants

of 2 ×2 matrices. The determinant is also related to the inverse of a matrix

[see Equations (6.60) and (6.61)]:

Theorem 7.3.1. The matrix A has an inverse if and only if det A =0in

which case

−1

det A

A =

det A

⎛

⎜

⎝

... A

⎞

⎟

⎠

, (7.13)

where A

is the cofactor of a

as deﬁned in Box 6.3.2.

7.3 The Determinant 223

Calculation of the determinant becomes extremely cumbersome when the

dimension of the matrix increases beyond 4 or 5. However, there are certain

properties of the determinant which may sometimes facilitate its calculation.

The determinant has the following properties:

some properties of

the determinant

1. To obtain the determinant of an n ×n matrix, multiply each element of

one row (or one column) by its cofactor and then add the results.

2. The determinant of the unit matrix is 1.

3. The determinant of a matrix is equal to the determinant of its transpose:

det A =detA

4. If two rows (or two columns) of a matrix are proportional (in particular,

equal), the determinant of the matrix is zero.

5. If a row or column—treated as a vector in R

—of a matrix is multiplied

by a constant, the determinant of the matrix will be multiplied by the

same constant.

6. If two rows (or two columns) of a matrix are interchanged, the determi-

nant changes sign.

7. The determinant will not change if we add to one row (or one column)

a multiple of another row (or another column). The addition of rows or

columns and their multiplication by numbers are to be understood as

operations in R

An important relation, which we state without proof,

is the determinant of

a product is the

product of

determinants.

det(AB)=detA det B. (7.14)

This, in combination with det 1 =1andAA

−1

= 1,gives

det(AA

−1

)=det1 ⇒ det A det(A

−1

)=1 ⇒ det(A

−1

det A

. (7.15)

In words, the determinant of the inverse of a matrix is the inverse of its

determinant of

inverse is inverse

determinant

determinant.

Recall that an orthogonal matrix A satisﬁes AA

= 1. The third property

of the determinant given above and (7.14) can be used to obtain

det(AA

)=det1 ⇒ (det A)

=1 ⇒ det A = ±1. (7.16)

Box 7.3.1. The determinant of an orthogonal matrix is either +1 or −1.

See Hassani, S. Mathematical Physics: A Modern Introduction to Its Foundations,

Springer-Verlag, 1999, Chapters 3 and 25.