数学专业英语(10) - 图文

Mathematical English 10: Dependent and Independent Sets Mathematical English Dr. Xiaomin Zhang Email: zhangxiaomin@nbu.edu.cn Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 1 Mathematical English 10: Dependent and Independent Sets §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 x1, …, xk, and corresponding set of scalars c1, …, ck, not all zero, such that ?cxi?1kii?0. The set S is called independent if it is not dependent. In this case, for all choices of distinct elements x1, …, xk in S and scalars c1, …, ck, ?cxi?1kii?0 implies c1=c2=?=ck=0 Although dependence and independence are properties of sets of Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 2 Mathematical English 10: Dependent and Independent Sets 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 nChapter 12 for the space Vn (i.e. R). 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 Vn Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 3 Mathematical English 10: Dependent and Independent Sets 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 u1(t) =cost, u2(t)=sint, u3(t)=1 for all real t. The Pythagorean identity shows that u1+u2-u3=0, so the three functions u1, u2, u3 are dependent. EXAMPLE 6 Let uk(t)=t for k=0,1 ,2, …, and t real. The set S={u0, u1, u2,…} is independent. To prove this , it suffices to show that for each n the n+1 polynomials u0, u1, …, un are independent. A relation of the form k22?cukk?0 means that kct(10.1) ?k?0 for all real t. When t=0, this gives c0=0. Differentiating (10.1) and setting t=0, we find that c1=0. Repeating the process, we find that each Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 4 Mathematical English 10: Dependent and Independent Sets coefficient ck is zero. EXAMPLE 7 If a1, …, an are distinct real numbers, the n exponential functions u1(x)=e, …, un(x)=ea1xanx 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 c1,…,cn such that ce?(10.2) kk?1nakx?0. Let aM be the largest of the n numbers a1,…,an. Multiplying both -axmembers of (10.2) by e, we obtain Mce?(10.3) kk?1n(ak?aM)x?0. 5 Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University

Mathematical English 10: Dependent and Independent Sets If k?M, the number ak?aM is negative. Therefore, when x?? in Equation (10.3), each term with k?M tends to zero and we find that cM=0. Deleting the Mth term from (10.2) and applying the induction hypothesis, we find that each of the remaining n?1 coefficients ck 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=Vn, 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 Vn is a linear space and not on any other special property of Vn. Therefore the proof given for Theorem 8.8 is valid for any linear space V. Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 6 Mathematical English 10: Dependent and Independent Sets 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 nEuclidean space R, 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. Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 7 Mathematical English 10: Dependent and Independent Sets 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 is called a real vector space, and C 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: nn2. 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: Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 8 Mathematical English 10: Dependent and Independent Sets 6. Distributivity of scalar sums: 7. Distributivity of vector sums: 8. Scalar multiplication identity: Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 9 Mathematical English 10: Dependent and Independent Sets 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. Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 10

Mathematical English 10: Dependent and Independent Sets DEFINITION If a linear space V has a basis of n elements, the integer n is called the dimension of V. We write n=dimV. if V={0}, we say V has dimension 0. EXAMPLE 1 The space Vn 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 2ndimension n+1. One basis is the set of n+1 polynomials {1, t, t, …, t}. 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 -x3xu1(x)=e, u2(x)=e. Every solution is a linear combination of these two. EXAMPLE 4 The space of all polynomials p(t) is infinite-dimensional. 2Although the infinite set {1, t, t,…} spans this space, no finite set of Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 11 Mathematical English 10: Dependent and Independent Sets polynomials spans the space. Another famous infinite-dimensional pspace is L-space. THEOREM 10.7 Let V be a finite-dimensional linear space with dimV=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, …, en are taken in a given order. We denote such an ordered basis as an n-tuple (e1, …, en). If x?V, we can express x as a linear combination of these basis elements: Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 12 Mathematical English 10: Dependent and Independent Sets nx?ce?ii. (10.4) i?1The coefficients in this equation determine an n-tuple of numbers (c1, … cn) that is uniquely determined by x. In fact, if we have another representation of x as a linear combination of e1, … ,en, say nx??diei, i?1n(ci?di)ei?0. But since the then by subtraction from (10.4), we find that ?i?1basis elements are independent, this implies ci=di, for each i, so we have (c1, …, cn)=(d1, …, dn). The components of the ordered n-tuple (c1, …, cn) determined by Equation (10.4) are called the components of x relative to the ordered basis (e1, …, en). Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 13 Mathematical English 10: Dependent and Independent Sets Notations L-space On a measure space X, the set of p-integrable functions is an pp2L-space (p>0). The set of L-functions generalizes L-space. Instead of square integrable, the measurable function f must be p-integrable for f pto be in L. On a measure space X, the L norm of a function f is pp The L-functions are the functions for which this integral converges. For pp?2, the space of L-functions is a Banach space which is not a Hilbert space. As in the case of an L-space, an L-function is really an equivalence class of functions which agree almost everywhere. Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 14 p2pMathematical English 10: Dependent and Independent Sets For p>1, the dual space to L is given by integrating against functions in qL, where 1/p+1/q=1. This makes sense because of H?lder's inequality p2for integrals. In particular, the only L-space which is self-dual is L. While the use of L-functions is not as common as L, they are very important in analysis and partial differential equations. For instance, p some operators are only bounded in Lfor some p>2. p2pDr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 15

Mathematical English 10: Dependent and Independent Sets 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, \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 pth Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 16 Mathematical English 10: Dependent and Independent Sets order.\Because Sylvester was interested in the determinant formed from the rectangular array of number and not the array itself, Sylvester used the term \which something else originates'. Sylvester (1851) subsequently used the term matrix informally, stating \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.\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, \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 Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 17 Mathematical English 10: Dependent and Independent Sets 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 Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 18 Mathematical English 10: Dependent and Independent Sets where the aij are called matrix elements. An m?n matrix consists of m rows and n columns, and the set of m?n m?nmatrices with real coefficients is sometimes denoted R. 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 aij=?ij Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 19 Mathematical English 10: Dependent and Independent Sets (where ?ij is the Kronecker delta) and the diagonal matrix aij=ci?ij (where ci 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 aij, and the matrix composed of entries aij 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 Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 20

Mathematical English 10: Dependent and Independent Sets on a single matrix are matrix diagonalization, matrix inversion, and transposition. The determinant det(A) or |A| of a matrix A is a very important quantity which appears in many diverse applications. The sum of the diagonal elements of a square matrix is known as the matrix trace Tr(A) and is also an important quantity in many sorts of computations. Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 21 Mathematical English 10: Dependent and Independent Sets Problem 1 If matrix A has m rows and l columns, B has n rows and m columns, C has k rows and n columns then matrix CBA has ____ rows and ____ columns. A k, l B m, n C n, l D k, m Problem 2 A square matrix An?n, n>3 is tridiagonal if aij=0 when i>j+1 or j>i+1, the other entries are unrestricted. In the following four matrices, which one is tridiagonal? ?1??3A ?0??0?224003350044???? B ????1??0?0??0?210032104??3?2? C ?1???1??2?3??0?012300120??0?0? D ?1???100???010???001? ?? Dr. Xiaomin Zhang: Mathematics Department, School of Science, Ningbo University 22

本文来源：https://www.bwwdw.com/article/x33r.html