Handbook of mathematics for engineers and scienteists part 33 ppsx

A real Euclidean space or simply, Euclidean space is a real linear space V endowed with a scalar product also called inner product, which is a real-valued function of two arguments xV, y

5.3.3-2 Relations between coordinate transformations and basis transformations.

Suppose that in a linear n-dimensional space V, the transition from its basis e1 , , e nto another basis2e1, ,2en is determined by the matrix A (see Paragraph 5.3.3-1) Let x be

any element of the spaceV with the coordinates (x1 , , x n) in the basis e1, , e nand the coordinates (2x1, , 2x n) in the basis2e1, ,2en, i.e.,

x = x1e1+· · · + x nen=2x12e1+· · · + 2x n2en Then using formulas (5.3.3.1), we obtain the following relations between these coordinates:

x j =

n



i=1

2x i a ij, 2x k =

n



l=1

x l b lk, j, k =1, , n.

In terms of matrices and row vectors, these relations can be written as follows:

(x1, , x n) = (2x1, , 2x n )A, (2x1, , 2x n ) = (x1, , x n )A–1

or, in terms of column vectors,

(x1, , x n)T = A T(2x1, , 2x n)T, (2x1, , 2x n)T = (A–1)T (x1, , x n)T,

where the superscript T indicates the transpose of a matrix.

5.4 Euclidean Spaces

5.4.1 Real Euclidean Space

5.4.1-1 Definition and properties of a real Euclidean space

A real Euclidean space (or simply, Euclidean space) is a real linear space V endowed with a

scalar product (also called inner product), which is a real-valued function of two arguments

xV, yV called the scalar product of these elements, denoted by x⋅y, and satisfying the

following conditions (axioms of the scalar product):

1 Symmetry: x⋅y = y⋅x.

2 Distributivity: (x1+ x2)⋅y = x1⋅y + x2⋅y.

3 Homogeneity: (λx)⋅y = λ(x⋅y) for any real λ.

4 Positive definiteness: x⋅x≥ 0for any x, and x⋅x =0if and only if x =0

If the nature of the elements and the scalar product is concretized, one obtains a specific

Euclidean space.

Example 1 Consider the linear space B3of all free vectors in three-dimensional space The space B3

becomes a Euclidean space if the scalar product is introduced as in analytic geometry (see Paragraph 4.5.3-1):

x⋅y =|x| |y|cos ϕ,

where ϕ is the angle between the vectors x and y.

Example 2 Consider the n-dimensional coordinate spaceRn whose elements are ordered systems of n

arbitrary real numbers, x = (x1, , x n) Endowing this space with the scalar product

x⋅y = x1y1+· · · + x n n,

we obtain a Euclidean space.

THEOREM For any two elements x and y of a Euclidean space, the Cauchy–Schwarz

inequality holds:

(x⋅y)2≤(x⋅x)(y⋅y).

A linear space V is called a normed space if it is endowed with a norm, which is a

real-valued function of xV, denoted by x and satisfying the following conditions:

1 Homogeneity: λx =|λ|x for any real λ.

2 Positive definiteness: x≥ 0andx =0if and only if x =0

3 The triangle inequality (also called the Minkowski inequality) holds for all elements

x and y:

x + y≤x + y. (5.4.1.1) The valuex is called the norm of an element x or its length.

THEOREM Any Euclidean space becomes a normed space if the norm is introduced by

COROLLARY In any Euclidean space with the norm (5.4.1.2), the triangle inequality

(5.4.1.1) holds for all its elements x and y.

The distance between elements x and y of a Euclidean space is defined by

d(x, y) = x – y.

One says that ϕ [0,2π] is the angle between two elements x and y of a Euclidean

space if

cos ϕ = x⋅y

x y. Two elements x and y of a Euclidean space are said to be orthogonal if their scalar product

is equal to zero, x⋅y =0

PYTHAGOREAN THEOREM Let x1, x m be mutually orthogonal elements of a

Eu-clidean space, i.e., xj⋅xj =0for i≠j Then

x1+· · · + x m 2 =x12+· · · + x m 2

Example 3 In the Euclidean space B3of free vectors with the usual scalar product (see Example 1), the following relations hold:

a =|a|, (a⋅b)2≤ |a| 2 |b| 2 , |a + b| ≤ |a| + |b|.

In the Euclidean space Rn of ordered systems of n numbers with the scalar product defined in Example 2,

the following relations hold:

x =x2+· · · + x2

n,

(x1y1+· · · + x n n)2≤(x2+· · · + x2n )(y2+· · · + y n2),

(x1+ y1 )2+· · · + (x n + y n)2≤x2+· · · + x2

n

y2+· · · + y2

n.

5.4.1-2 Orthonormal basis in a finite-dimensional Euclidean space

For elements x1, , x m of a Euclidean space, the mth-order determinant det[x i ⋅xj] is

called their Gram determinant These elements are linearly independent if and only if their

Gram determinant is different from zero

One says that n elements i1, , i n of an n-dimensional Euclidean space V form its

orthonormal basis if these elements have unit norm and are mutually orthogonal, i.e.,

ii⋅i =1 for i = j,

0 for i≠j.

THEOREM In any n-dimensional Euclidean space V, there exists an orthonormal basis.

Orthogonalization of linearly independent elements:

Let e1, , e n be n linearly independent vectors of an n-dimensional Euclidean space V.

From these vectors, one can construct an orthonormal basis of V using the following algorithm (called Gram–Schmidt orthogonalization):

ii = √gi

gi⋅gi, where gi = ei–

i



j=1

(ei⋅i )ij (i =1,2, , n). (5.4.1.3)

Remark. In any n-dimensional (n >1) Euclidean spaceV, there exist infinitely many orthonormal bases.

Properties of an orthonormal basis of a Euclidean space:

1 Let i1, , i nbe an orthonormal basis of a Euclidean spaceV Then the scalar product

of two elements x = x1i1+· · · + x nin and y = y1i1+· · · + y ninis equal to the sum of products of their respective coordinates:

x⋅y = x1y1+· · · + x n y n.

2 The coordinates of any vector x in an orthonormal basis i1, , i nare equal to the scalar

product of x and the corresponding vector of the basis (or the projection of the element

x on the axis in the direction of the corresponding vector of the basis):

x k= x⋅ik (k =1,2, , n).

Remark. In an arbitrary basis e1, , en of a Euclidean space, the scalar product of two elements

x = x1e1 +· · · + x nen and y = y1e1 +· · · + y nenhas the form

x⋅y =

n



i=1

n



j=1

a ij x i y j,

where a ij= ei⋅ej (i, j =1, 2, , n).

LetX , Y be subspaces of a Euclidean space V The subspace X is called the orthogonal

complement of the subspace Y in V if any element x of X is orthogonal to any element y of Y

andX⊕Y = V.

THEOREM Any n-dimensional Euclidean space V can be represented as the direct sum

of its arbitrary subspaceY and its orthogonal complement X

Two Euclidean spacesV and 2V are said to be isomorphic if one can establish a one-to-one

correspondence between the elements of these spaces satisfying the following conditions:

if elements x and y of V correspond to elements 2x and 2y of 2V, then the element x + y

corresponds to2x+2y; the element λx corresponds to λ2x for any λ; the scalar product (x⋅y)V

is equal to the scalar product (2x⋅2y)2V.

THEOREM Any two n-dimensional Euclidean spaces V and 2V are isomorphic.

5.4.2 Complex Euclidean Space (Unitary Space)

5.4.2-1 Definition and properties of complex Euclidean space (unitary space)

A complex Euclidean space (or unitary space) is a complex linear space V endowed with

a scalar product (also called inner product), which is a complex-valued function of two

arguments x  V and y  V called their scalar product, denoted by x⋅y, satisfying the

following conditions (called axioms of the scalar product):

1 Commutativity: x⋅y = y⋅x.

2 Distributivity: (x1+ x2)⋅y = x1⋅y + x2⋅y.

3 Homogeneity: (λx)⋅y = λ(x⋅y) for any complex λ.

4 Positive definiteness: x⋅x≥ 0; and x⋅x =0if and only if x =0

Here y⋅x is the complex conjugate of a number y⋅x.

Example 1 Consider the n-dimensional complex linear spaceRn

∗ whose elements are ordered systems

of n complex numbers, x = (x1, , x n) We obtain a unitary space if the scalar product of two elements

x = (x1, , x n ) and y = (y1, , y n) is introduced by

x⋅y = x1¯y1 +· · · + x n ¯yn, where¯yj is the complex conjugate of y j.

THEOREM For any two elements x and y of an arbitrary unitary space, the Cauchy–

Schwarz inequality holds:

|x⋅y|2≤(x⋅x)(y⋅y).

THEOREM Any unitary space becomes a normed space if the norm of its element x is

introduced by

COROLLARY For any two elements x and y of a normed Euclidean space with the norm

(5.4.2.1), the triangle inequality (5.4.1.1) holds

The distance between elements x and y of a unitary space is defined by

Two elements x and y of a unitary space are said to be orthogonal if their scalar product

is equal to zero, x⋅y =0

5.4.2-2 Orthonormal basis in a finite-dimensional unitary space

Elements x1, , x mof a unitary spaceV are linearly independent if and only if their Gram

determinant is different from zero, det[x i⋅xj ≠ 0

One says that elements i1, , i n of an n-dimensional unitary space V form an

or-thonormal basis of that space if these elements are mutually orthogonal and have unit norm,

i.e.,

ii⋅i =

1 for i = j,

0 for i≠j.

Given any n linearly independent elements of a unitary space, one can construct an

orthonormal basis of that space using the procedure described in Paragraph 5.4.1-2 (see formulas (5.4.1.3))

Properties of an orthonormal basis of a unitary space:

1 Let i1, , i nbe an orthonormal basis in a unitary space Then the scalar product of

two elements x = x1i1+· · · + x nin and y = y1i1+· · · + y ninis equal to the sum

x⋅y = x1¯y1+· · · + x n ¯y n.

2 The coordinates of any vector in an orthonormal basis i1, , inare equal to the scalar products of this vector and the vectors of the bases (or the projections of this element

on the axes in the direction of the corresponding basis vectors):

x k= x⋅ik (k =1, 2, , n).

Two unitary spacesV and 2V are said to be isomorphic if there is a one-to-one

corre-spondence between their elements satisfying the following conditions: if elements x and

y ofV correspond to elements 2x and 2y of 2V, then x + y corresponds to 2x + 2y; the element

λx corresponds to λ 2x for any complex λ; the scalar product (x⋅y)V is equal to the scalar product (2x⋅2y)2V.

THEOREM Any two n-dimensional unitary spaces V and 2V are isomorphic.

5.4.3 Banach Spaces and Hilbert Spaces

5.4.3-1 Convergence in unitary spaces Banach space

Any normed linear space is a metric space with the metric (5.4.2.2).

A sequence{bs}of elements of a normed spaceV is said to be convergent to an element

bV as s → ∞ if lim

s→∞ b s– b =0

A series x0+ x1+· · · with terms in a normed space is said to be convergent to an element

x (called its sum; one writes x = lim

n→∞

n



k=0xk =

∞



k=0xk) if the sequence of its partial sums

forms a sequence convergent to x, i.e., lim

n→∞

33

3x –n

k=0xk

33

3 =0

A normed linear space V is said to be complete if any sequence of its elements s0,

s , satisfying the condition

lim

n,m→∞ s n– sm  =0

is convergent to some element s of the spaceV.

A complete normed linear space is called a Banach space.

Remark Any finite-dimensional normed linear space is complete.

5.4.3-2 Hilbert space

A complete unitary space is called a Hilbert space.

Any complete subspace of a Hilbert space is itself a Hilbert space

PROJECTION THEOREM LetV1be a complete subspace of a unitary spaceV Then for

any xV, there is a unique vector x p V1such that

min

y V1x – y = x – x p .

Moreover, the vector xp is the unique element of V1 for which the difference x – xp is

orthogonal to any element x1ofV1, i.e.,

(x – xp)⋅x1=0 for all x1V1

The mapping x→ x p is a bounded linear operator fromV to V1 called the orthogonal

projection of the space V to its subspace V1

5.5 Systems of Linear Algebraic Equations

5.5.1 Consistency Condition for a Linear System

5.5.1-1 Notion of a system of linear algebraic equations

A system of m linear equations with n unknown quantities has the form

a11x1+ a12x2+· · · + a1 k x k+· · · + a1 n x n = b1,

a21x1+ a22x2+· · · + a2 k x k+· · · + a2 n x n = b2,

a m1x1+ a m2x2+· · · + a mk x k+· · · + a mn x n = b m,

(5.5.1.1)

where a11, a12, , a mn are the coefficients of the system; b1, b2, , b m are its free terms; and x1, x2, , x nare the unknown quantities

System (5.5.1.1) is said to be homogeneous if all its free terms are equal to zero Other-wise (i.e., if there is at least one nonzero free term) the system is called nonhomogeneous.

If the number of equations is equal to that of the unknown quantities (m = n), sys-tem (5.5.1.1) is called a square syssys-tem.

A solution of system (5.5.1.1) is a set of n numbers x1, x2, , x n satisfying the

equations of the system A system is said to be consistent if it admits at least one solution.

If a system has no solutions, it is said to be inconsistent A consistent system of the form (5.5.1.1) is called a determined system—it has a unique solution A consistent system with more than one solution is said to be underdetermined.

It is convenient to use matrix notation for systems of the form (5.5.1.1),

where A≡[a ij ] is a matrix of size m×n called the basic matrix of the system; X ≡[x i] is

a column vector of size n; B ≡[b i ] is a column vector of size m.

5.5.1-2 Existence of nontrivial solutions of a homogeneous system

Consider a homogeneous system

where A≡ [a ij ] is its basic matrix of size m×n, X ≡ [x i ] is a column vector of size n, and O m≡[0] is a column vector of size m System (5.5.1.3) is always consistent since it always has the so-called trivial solution X≡O n.

THEOREM A homogeneous system (5.5.1.3) has a nontrivial solution if and only if the

rank of the matrix A is less than the number of the unknown quantities n.

It follows that a square homogeneous system has a nontrivial solution if and only if the

determinant of its matrix of coefficients is equal to zero, det A =0

Properties of the set of all solutions of a homogeneous system:

1 All solutions of a homogeneous system (5.5.1.3) form a linear space

2 The linear space of all solutions of a homogeneous system (5.5.1.3) with n unknown quantities and a basic matrix of rank r is isomorphic to the space A n–r of all ordered

systems of (n – r) numbers The dimension of the space of solutions is equal to n – r.

3 Any system of (n–r) linearly independent solutions of the homogeneous system (5.5.1.3) forms a basis in the space of all its solutions and is called a fundamental system of

solutions of that system The fundamental system of solutions corresponding to the

basis i1 = (1,0, ,0), i2 = (0,1, ,0), , i n–r = (0,0, ,1) of the space A n–r is

said to be normal.

5.5.1-3 Consistency condition for a general linear system

System (5.5.1.1) or (5.5.1.2) is associated with two matrices: the basic matrix A of size

m×n and the augmented matrix A1of size m×(n+1) formed by the matrix A supplemented

with the column of the free terms, i.e.,

A1≡

⎛

⎜

⎜

⎝

a11 a12 · · · a1 n b1

a21 a22 · · · a2 n b2

. . .

a m1 a m2 · · · a mn b m

⎞

⎟

⎟

⎠ (5.5.1.4)

KRONECKER–CAPELLI THEOREM A linear system (5.5.1.1) [or (5.5.1.2)] is consistent

if and only if its basic matrix and its augmented matrix (5.5.1.4) have the same rank, i.e

rank (A1) = rank (A).

5.5.2 Finding Solutions of a System of Linear Equations

5.5.2-1 System of two equations with two unknown quantities

A system of two equations with two unknown quantities has the form

a1x + b1y = c1,

a2x + b2y = c2 (5.5.2.1)

Depending on the coefficients a k , b k , c k, the following three cases are possible:

1◦ IfΔ = a1b2– a2b1≠ 0, then system (5.5.2.1) has a unique solution,

x= c1b2– c2b1

a1b2– a2b1, y =

a1c2– a2c1

a1b2– a2b1.

2◦ IfΔ = a1b2– a2b1=0and a1c2– a2c1=0(the case of proportional coefficients), then system (5.5.2.1) has infinitely many solutions described by the formulas

x = t, y= c1– a1t

b1 (b1≠ 0),

where t is arbitrary.

3◦ IfΔ = a1b2– a2b1=0and a1c2– a2c1 ≠ 0, then system (5.5.2.1) has no solutions

solutions of. .. unknown quantities and a basic matrix of rank r is isomorphic to the space A n–r of all ordered

systems of (n – r) numbers The dimension of the space of solutions is equal... convenient to use matrix notation for systems of the form (5.5.1.1),

where A≡[a ij ] is a matrix of size m×n called the basic matrix of the system; X ≡[x i]