数学专业英语(10)

Mathematical English Dr. Xiaomin Zhang Email: zhangxiaomin@https://www.360docs.net/doc/7d3997415.html,

§2.10 Dependent and Independent Sets

TEXT A Dependent and independent sets in a linear space

DEFINITION A set of S of elements in a linear space V is called dependent if there is a finite set of distinct elements in S, say x 1, …, x k, and corresponding set of scalars c 1, …, c k , not all zero, such that

10.k i i i c x

==∑

The set S is called independent if it is not dependent. In this case, for all choices of distinct elements x 1, …, x k in S and scalars c 1, …, c k ,

10k i i i c x

==∑ implies c 1=c 2=?=c k =0

Although dependence and independence are properties of sets of

elements, we also apply these terms to the elements themselves. For example, the elements in an independent set are called independent elements.

If S is a finite set, the foregoing definition agrees with that given in Chapter 12 for the space V n (i.e. R n). However, the present definition is not restricted to finite sets.

EXAMPLE 1 If a subset T of a set S is dependent, then S itself is dependent. This is logically equivalent to the statement that every subset of an independent set is independent.

EXAMPLE 2 If one element in S is a scalar multiple of another, then S is dependent.

EXAMPLE 3 If 0 S, then S dependent.

EXAMPLE 4 The empty set is independent.

Many examples of dependent and independent sets of vectors in V n

were discussed in Chapter 8. The following examples illustrate these concepts in function spaces. In each case the underlying linear space V is the set of all real-valued functions defined on the real line.

EXAMPLE 5 Let u 1(t) =cos 2t, u 2(t)=sin 2t, u 3(t)=1 for all real t . The Pythagorean identity shows that u 1+u 2-u 3=0, so the three functions u 1, u 2, u 3 are dependent.

EXAMPLE 6 Let u k (t)=t k for k=0,1 ,2, …, and t real. The set S={u 0, u 1, u 2,…} is independent. To prove this , it suffices to show that for each n the n+1 polynomials u 0, u 1, …, u n are independent. A relation of the form 0k k c u =∑ means that

(10.1) 0k k c t =∑

for all real t . When t=0, this gives c 0=0. Differentiating (10.1) and setting t=0, we find that c 1=0. Repeating the process, we find that each

coefficient c k is zero.

EXAMPLE 7 If a 1, …, a n are distinct real numbers, the n exponential functions

u 1(x)=e a 1x , …, u n (x)=e a n x

are independent. We can prove this by induction on n . The result holds trivially when n=1. Therefore, assume it is true for n-1 exponential functions and consider scalars c 1,…,c n such that

(10.2) 10a x k n k

k c e ==∑.

Let a M be the largest of the n numbers a 1,…,a n . Multiplying both

members of (10.2) by e -a M

x , we obtain (10.3) ()10k M n a a x k

k c e -==∑.

If k≠M, the number a k-a M is negative. Therefore, when x→∞ in Equation (10.3), each term with k≠M tends to zero and we find that c M=0. Deleting the M th term from (10.2) and applying the induction hypothesis, we find that each of the remaining n-1 coefficients c k is zero.

THEOREM 10.5 Let S be an independent set consisting of k elements in a linear space V and let L(S) be the subspace spanned by S. Then every set of k+1 elements in L(S) is dependent.

Proof. When V=V n, Theorem 10.5 reduces to Theorem 8.8. If we examine the proof of Theorem 8.8, we find that it is based only on the fact that V n is a linear space and not on any other special property of V n. Therefore the proof given for Theorem 8.8 is valid for any linear space V.

Notations

linear space The concept of a space is an extremely general and important mathematical construct. Members of the space obey certain addition properties. Spaces which have been investigated and found to be of interest are usually named after one or more of their investigators. This practice unfortunately leads to names which give very little insight into the relevant properties of a given space. One of the most general type of mathematical spaces is the topological space.

linear space, also vector space, is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space R n, where every element is represented by a list of n real numbers, scalars are real numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.

For a general vector space, the scalars are members of a field F, in which case V is called a vector space over F.

Euclidean n-space R n is called a real vector space, and C n is called a complex vector space.

In order for V to be a vector space, the following conditions must hold for all elements X, Y, Z∈V and any scalars r, s∈F:

1. Commutativity:

2. Associativity of vector addition:

3. Additive identity: For all X,

4. Existence of additive inverse: For any X, there exists a -X such that

5. Associativity of scalar multiplication:

6. Distributivity of scalar sums:

7. Distributivity of vector sums:

8. Scalar multiplication identity:

TEXT B Basis and dimension

DEFINITION A finite set S of elements in a linear space V is called a finite basis for V if S is independent and spans V. The space V is called finite dimensional if it has a finite basis, or if V consists of 0 alone. Otherwise V is called infinite dimensional.

THEOREM 10.6 Let V be a finite-dimensional linear space. Then every finite basis for V has the same number of elements.

Proof Let S and T be two finite bases for V. Suppose S consists of k elements and T consists of m elements. Since S is independent and spans V, Theorem 10.5 tells us that every set of k+1 elements in V is dependent. Therefore, every set of more than k elements in V is dependent. Since T is an independent set, we must have m≤k. The same argument with S and T interchanged shows that k≤m. Therefore k=m.

DEFINITION If a linear space V has a basis of n elements, the integer n is called the dimension of V. We write n=dim V. if V={0}, we say V has dimension 0.

EXAMPLE 1 The space V n has dimension n. One basis is the set of n unit coordinate vectors.

EXAMPLE 2 The space of all polynomials p(t)of degree ≤n has dimension n+1. One basis is the set of n+1 polynomials {1, t, t2, …, t n}. Every polynomial of degree ≤n is a linear combination of these n+1 polynomials.

EXAMPLE 3 The space of solutions of the differential equation y"-2y'-3y=0 has dimension 2. One basis consists of the two functions u1(x)=e-x, u2(x)=e3x. Every solution is a linear combination of these two.

EXAMPLE 4 The space of all polynomials p(t)is infinite-dimensional. Although the infinite set {1, t, t2,…}spans this space, no finite set of

polynomials spans the space. Another famous infinite-dimensional space is L p-space.

THEOREM 10.7 Let V be a finite-dimensional linear space with dim V=n. Then we have the following.

(a) Any set of independent elements in V is a subset of some basis for

V.

(b) Any set of n independent elements is a basis for V.

Proof. The proof of (a) is identical to that of part(b) of Theorem 8.10. The proof of (b) is identical to that of part (c) of Theorem 8.10.

Let V be a linear space of dimension n and consider a basis whose elements e1, …, e n are taken in a given order. We denote such an ordered basis as an n-tuple (e1, …, e n). If x V, we can express x as a linear combination of these basis elements:

(10.4) 1

n

i i i x c e ==∑. The coefficients in this equation determine an n -tuple of numbers (c 1, … c n ) that is uniquely determined by x . In fact, if we have another representation of x as a linear combination of e 1, … ,e n , say

1n

i i i x d e ==∑, then by subtraction from (10.4), we find that 1

()0n i i i i c d e =-=∑. But since the basis elements are independent, this implies c i =d i , for each i , so we have (c 1, …, c n )=(d 1, …, d n ).

The components of the ordered n -tuple (c 1, …, c n ) determined by Equation (10.4) are called the components of x relative to the ordered basis (e 1, …, e n ).

Notations

L p-space On a measure space X, the set of p-integrable functions is an L p-space (p>0). The set of L p-functions generalizes L2-space. Instead of square integrable, the measurable function f must be p-integrable for f to be in L p.

On a measure space X, the L p norm of a function f is

The L p-functions are the functions for which this integral converges. For p 2, the space of L p-functions is a Banach space which is not a Hilbert space.

As in the case of an L2-space, an L p-function is really an equivalence class of functions which agree almost everywhere.

For p>1, the dual space to L p is given by integrating against functions in L q, where 1/p+1/q=1. This makes sense because of H?lder's inequality for integrals. In particular, the only L p-space which is self-dual is L2.

While the use of L p-functions is not as common as L2, they are very important in analysis and partial differential equations. For instance, some operators are only bounded in L p for some p>2.

SUPPLEMENT Matrix

A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, for every linear transformation, there exists exactly one corresponding matrix, and every matrix corresponds to a unique linear transformation. The matrix, and its close relative the determinant, are extremely important concepts in linear algebra, and were first formulated by Sylvester (1850) and Cayley.

In his 1850 paper, Sylvester wrote, "For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding of p th

order." Because Sylvester was interested in the determinant formed from the rectangular array of number and not the array itself, Sylvester used the term "matrix" in its conventional usage to mean 'the place from which something else originates'. Sylvester (1851) subsequently used the term matrix informally, stating "Form the rectangular matrix consisting of n rows and (n+1) columns.... Then all the n+1 determinants that can be formed by rejecting any one column at pleasure out of this matrix are identically zero." However, it remained up to Sylvester's collaborator Cayley to use the terminology in its modern form in papers of 1855 and 1858.

In his 1867 treatise on determinants, C.L. Dodgson objected to the use of the term 'matrix,' stating, "I am aware that the word 'Matrix' is already in use to express the very meaning for which I use the word 'Block'; but surely the former word means rather the mould, or form, into which algebraical quantities may be introduced, than an actual assemblage of

such quantities...." However, Dodgson's objections have passed unheeded and the term 'matrix' has stuck.

The transformation given by the system of equations

is represented as a matrix equation by

where the a ij are called matrix elements.

An m?n matrix consists of m rows and n columns, and the set of m?n matrices with real coefficients is sometimes denoted R m?n. To remember which index refers to which direction, identify the indices of the last (i.e., lower right) term, so the indices m, n of the last element in the above matrix identifies it as an m?n matrix.

A matrix is said to be square if m=n, and rectangular if m≠n. An m?1 matrix is called a column vector, and a 1?n matrix is called a row vector. Special types of square matrices include the identity matrix I, with a ij=δij

(where δij is the Kronecker delta) and the diagonal matrix a ij=c iδij(where c i are a set of constants).

In this work, matrices are represented using square brackets as delimiters, but in the general literature, they are more commonly delimited using parentheses.

The transformation given in the above equation can be written X'=AX, where X'and X are vectors and A is a matrix.

It is sometimes convenient to represent an entire matrix in terms of its matrix elements. Therefore, the (i, j)th element of the matrix A could be written a ij, and the matrix composed of entries a ij could be written as (a)ij for short.

Two matrices may be added (matrix addition) or multiplied (matrix multiplication) together to yield a new matrix. Other common operations