线性代数 英文讲义
线性代数 英文讲义
Chapter 1 Matrices and Systems of EquationsLinear systems arise in applications to such areas as engineering, physics, electronics, business, economics, sociology（社会学）, ecology （生态学）, demography(人口统计学), and genetics（遗传学）, etc. §1. Systems of Linear EquationsNew words and phrases in this section:Linear equation 线性方程Linear system，System of linear equations 线性方程组Unknown 未知量Consistent 相容的Consistence 相容性Inconsistent不相容的Inconsistence 不相容性Solution 解Solution set 解集Equivalent 等价的Equivalence 等价性Equivalent system 等价方程组Strict triangular system 严格上三角方程组Strict triangular form 严格上三角形式Back Substitution 回代法Matrix 矩阵Coefficient matrix 系数矩阵Augmented matrix 增广矩阵Pivot element 主元Pivotal row 主行Echelon form 阶梯形1.1 DefinitionsA linear equation (线性方程) in n unknowns（未知量）is1122...n na x a x a x b+++=A linear system of m equations in n unknowns is11112211211222221122...... .........n n n n m m m n n m a x a x a x b a x a x a x b a x a x a x b+++=⎧⎪+++=⎪⎨⎪⎪+++=⎩ This is called a m x n (read as m by n) system.A solution to an m x n system is an ordered n-tuple of numbers （n 元数组）12(,,...,)n x x x that satisfies all the equations.A system is said to be inconsistent （不相容的） if the system has no solutions.A system is said to be consistent （相容的）if the system has at least one solution.The set of all solutions to a linear system is called the solution set（解集）of the linear system.1.2 Geometric Interpretations of 2x2 Systems11112212112222a x a xb a x a x b +=⎧⎨+=⎩ Each equation can be represented graphically as a line in the plane. The ordered pair 12(,)x x will be a solution if and only if it lies on bothlines.In the plane, the possible relative positions are(1) two lines intersect at exactly a point; (The solution set has exactly one element)(2)two lines are parallel; (The solution set is empty)(3)two lines coincide. (The solution set has infinitely manyelements)The situation is the same for mxn systems. An mxn system may not be consistent. If it is consistent, it must either have exactly one solution or infinitely many solutions. These are only possibilities.Of more immediate concerns is the problem of finding all solutions to a given system.1.3 Equivalent systemsTwo systems of equations involving the same variables are said to be equivalent（等价的，同解的）if they have the same solution set.To find the solution set of a system, we usually use operations to reduce the original system to a simpler equivalent system.It is clear that the following three operations do not change the solution set of a system.(1)Interchange the order in which two equations of a system arewritten;(2)Multiply through one equation of a system by a nonzero realnumber;(3)Add a multiple of one equation to another equation. (subtracta multiple of one equation from another one)Remark: The three operations above are very important in dealing with linear systems. They coincide with the three row operations of matrices. Ask a student about the proof.1.4 n x n systemsIf an nxn system has exactly one solution, then operation 1 and 3 can be used to obtain an equivalent “strictly triangular system ”A system is said to be in strict triangular form (严格三角形) if in the k-th equation the coefficients of the first k-1 variables are all zero and the coefficient ofkx is nonzero. (k=1, 2, …,n)An example of a system in strict triangular form:123233331 2 24x x x x x x ++=⎧⎪-=⎨⎪=⎩Any nxn strictly triangular system can be solved by back substitution (回代法).(Note: A phrase: “substitute 3 for x ” == “replace x by 3”)In general, given a system of linear equations in n unknowns, we will use operation I and III to try to obtain an equivalent system that is strictly triangular.We can associate with a linear system an mxn array of numbers whose entries are coefficient of theix ’s. we will refer to this array as thecoefficient matrix (系数矩阵) of the system.111212122212.....................n nm m m n a a a a a a a a a ⎛⎫⎪ ⎪ ⎪ ⎪⎝⎭A matrix （矩阵） is a rectangular array of numbersIf we attach to the coefficient matrix an additional column whose entries are the numbers on the right-hand side of the system, we obtain the new matrix11121121222212n n s m m m na a ab a a a b b a a a ⎛⎫ ⎪ ⎪ ⎪⎝⎭We refer to this new matrix as the augmented matrix （增广矩阵） of a linear system.The system can be solved by performing operations on the augmented matrix. i x ’s are placeholders that can be omitted until the endof computation.Corresponding to the three operations used to obtain equivalent systems, the following row operation may be applied to the augmented matrix.1.5 Elementary row operationsThere are three elementary row operations:(1)Interchange two rows;(2)Multiply a row by a nonzero number;(3)Replace a row by its sum with a multiple of another row.Remark: The importance of these three operations is that they do not change the solution set of a linear system and may reduce a linear system to a simpler form.An example is given here to illustrate how to perform row operations on a matrix.★Example:The procedure for applying the three elementary row operations:Step 1: Choose a pivot element (主元)(nonzero) from among the entries in the first column. The row containing the pivotnumber is called a pivotal row（主行）. We interchange therows (if necessary) so that the pivotal row is the new firstrow.Multiples of the pivotal row are then subtracted form each of the remaining n-1 rows so as to obtain 0’s in the firstentries of rows 2 through n.Step2: Choose a pivot element from the nonzero entries in column 2, rows 2 through n of the matrix. The row containing thepivot element is then interchanged with the second row ( ifnecessary) of the matrix and is used as the new pivotal row.Multiples of the pivotal row are then subtracted form eachof the remaining n-2 rows so as to eliminate all entries belowthe pivot element in the second column.Step 3: The same procedure is repeated for columns 3 through n-1.Note that at the second step, row 1 and column 1 remain unchanged, at the third step, the first two rows and first two columns remain unchanged, and so on.At each step, the overall dimensions of the system are effectively reduced by 1. (The number of equations and the number of unknowns all decrease by 1.)If the elimination process can be carried out as described, we will arrive at an equivalent strictly triangular system after n-1 steps.However, the procedure will break down if all possible choices for a pivot element are all zero. When this happens, the alternative is to reduce the system to certain special echelon form（梯形矩阵）. AssignmentStudents should be able to do all problems.Hand-in problems are: # 7--#11§2. Row Echelon FormNew words and phrases:Row echelon form 行阶梯形Reduced echelon form 简化阶梯形 Lead variable 首变量 Free variable 自由变量Gaussian elimination 高斯消元Gaussian-Jordan reduction. 高斯-若当消元 Overdetermined system 超定方程组 Underdetermined systemHomogeneous system 齐次方程组 Trivial solution 平凡解2.1 Examples and DefinitionIn this section, we discuss how to use elementary row operations to solve mxn systems.Use an example to illustrate the idea.★ Example : Example 1 on page 13. Consider a system represented by the augmented matrix111111110011220031001131112241⎛⎫ ⎪--- ⎪ ⎪-- ⎪- ⎪ ⎪⎝⎭ 111111001120002253001131001130⎛⎫⎪ ⎪ ⎪ ⎪- ⎪ ⎪⎝⎭………..(The details will given in class)We see that at this stage the reduction to strict triangular form breaks down. Since our goal is to simplify the system as much as possible, we move over to the third column. From the example above, we see that the coefficient matrix that we end up with is not in strict triangular form,it is in staircase or echelon form (梯形矩阵).111111001120000013000004003⎛⎫ ⎪ ⎪ ⎪ ⎪- ⎪ ⎪-⎝⎭The equations represented by the last two rows are:12345345512=0 2=3 0=4 03x x x x x x x x x ++++=⎧⎪++⎪⎪⎨⎪-⎪=-⎪⎩Since there are no 5-tuples that could possibly satisfy these equations, the system is inconsistent.Change the system above to a consistent system.111111110011220031001133112244⎛⎫ ⎪--- ⎪ ⎪-- ⎪ ⎪ ⎪⎝⎭ 111111001120000013000000000⎛⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭The last two equations of the reduced system will be satisfied for any 5-tuple. Thus the solution set will be the set of all 5-tuples satisfying the first 3 equations.The variables corresponding to the first nonzero element in each row of the augment matrix will be referred to as lead variable .（首变量） The remaining variables corresponding to the columns skipped in the reduction process will be referred to as free variables （自由变量）.If we transfer the free variables over to the right-hand side in the above system, then we obtain the system:1352435451 2 3x x x x x x x x x ++=--⎧⎪+=-⎨⎪=⎩which is strictly triangular in the unknown 1x 3x 5x . Thus for each pairof values assigned to 2xand4x , there will be a unique solution.★Definition: A matrix is said to be in row echelon form (i) If the first nonzero entry in each nonzero row is 1.(ii)If row k does not consist entirely of zeros, the number of leading zero entries in row k+1 is greater than the number of leading zero entries in row k.(iii) If there are rows whose entries are all zero, they are below therows having nonzero entries.★Definition : The process of using row operations I, II and III to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination （高斯消元法）.Note that row operation II is necessary in order to scale the rows so that the lead coefficients are all 1.It is clear that if the row echelon form of the augmented matrix contains a row of the form (), the system is inconsistent.000|1Otherwise, the system will be consistent.If the system is consistent and the nonzero rows of the row echelon form of the matrix form a strictly triangular system (the number of nonzero rows<the number of unknowns), the system will have a unique solution. If the number of nonzero rows<the number of unknowns, then the system has infinitely many solutions. (There must be at least one free variable. We can assign the free variables arbitrary values and solve for the lead variables.)2.2 Overdetermined SystemsA linear system is said to be overdetermined if there are more equations than unknowns.2.3 Underdetermined SystemsA system of m linear equations in n unknowns is said to be underdetermined if there are fewer equations than unknowns (m<n). It is impossible for an underdetermined system to have only one solution.In the case where the row echelon form of a consistent system has free variables, it is convenient to continue the elimination process until all the entries above each lead 1 have been eliminated. The resulting reduced matrix is said to be in reduced row echelon form. For instance,111111001120000013000000000⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭ 110004001106000013000000000⎛⎫⎪- ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭Put the free variables on the right-hand side, it follows that12345463x x x x x =-=--=Thus for any real numbersαandβ, the 5-tuple()463ααββ---is a solution.Thus all ordered 5-tuple of the form ()463ααββ--- aresolutions to the system.2.4 Reduced Row Echelon Form★Definition : A matrix is said to be in reduced row echelon form if :(i)the matrix is in row echelon form.(ii) The first nonzero entry in each row is the only nonzero entry in its column.The process of using elementary row operations to transform a matrix into reduced echelon form is called Gaussian-Jordan reduction.The procedure for solving a linear system:(i) Write down the augmented matrix associated to the system; (ii) Perform elementary row operations to reduce the augmented matrix into a row echelon form;(iii) If the system if consistent, reduce the row echelon form into areduced row echelon form. (iv) Write the solution in an n-tuple formRemark: Make sure that the students know the difference between the row echelon form and the reduced echelon form.Example 6 on page 18: Use Gauss-Jordan reduction to solve the system:1234123412343030220x x x x x x x x x x x x -+-+=⎧⎪+--=⎨⎪---=⎩The details of the solution will be given in class.2.5 Homogeneous SystemsA system of linear equations is said to be homogeneous if theconstants on the right-hand side are all zero.Homogeneous systems are always consistent since it has a trivial solution. If a homogeneous system has a unique solution, it must be the trivial solution.In the case that m<n (an underdetermined system), there will always free variables and, consequently, additional nontrivial solution.Theorem 1.2.1 An mxn homogeneous system of linear equations has a nontrivial solution if m<n.Proof A homogeneous system is always consistent. The row echelon form of the augmented matrix can have at most m nonzero rows. Thus there are at most m lead variables. There must be some free variable. The free variables can be assigned arbitrary values. For each assignment of values to the free variables, there is a solution to the system.AssignmentStudents should be able to do all problems except 17, 18, 20.Hand-in problems are 9, 10, 16,Select one problem from 14 and 19.§3. Matrix AlgebraNew words and phrases:Algebra 代数Scalar 数量，标量Scalar multiplication 数乘 Real number 实数 Complex number 复数 V ector 向量Row vector 行向量 Column vector 列向量Euclidean n-space n 维欧氏空间 Linear combination 线性组合 Zero matrix 零矩阵Identity matrix 单位矩阵 Diagonal matrix 对角矩阵 Triangular matrix 三角矩阵Upper triangular matrix 上三角矩阵 Lower triangular matrix 下三角矩阵 Transpose of a matrix 矩阵的转置(Multiplicative ) Inverse of a matrix 矩阵的逆 Singular matrix 奇异矩阵 Singularity 奇异性Nonsingular matrix 非奇异矩阵 Nonsingularity 非奇异性The term scalar (标量，数量) is referred to as a real number (实数) or a complex number （复数）. Matrix notationAn mxn matrix, a rectangular array of mn numbers.111212122212.....................n nm m m n a a a a a a a a a ⎛⎫⎪ ⎪ ⎪ ⎪⎝⎭()ij A a =3.1 VectorsMatrices that have only one row or one column are of special interest since they are used to represent solutions to linear systems.We will refer to an ordered n-tuple of real numbers as a vector （向量）.If an n-tuple is represented in terms of a 1xn matrix, then we will refer to it as a row vector . Alternatively, if the n-tuple is represented by an nx1 matrix, then we will refer to it as a column vector . In this course, we represent a vector as a column vector.The set of all nx1 matrices of real number is called Euclidean n-space (n 维欧氏空间) and is usually denoted by nR.Given a mxn matrix A, it is often necessary to refer to a particular row or column. The matrix A can be represented in terms of either its column vectors or its row vectors.12(a ,a ,,a )n A = ora (1,:)a(2,:)a(,:)A m ⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭3.2 EqualityFor two matrices to be equal, they must have the same dimensions and their corresponding entries must agree★Definition : Two mxn matrices A and B are said to be equal ifij ij a b =for each ordered pair (i, j)3.3 Scalar MultiplicationIf A is a matrix,αis a scalar, thenαA is the mxn matrix formed by multiplying each of the entries of A byα.★Definition : If A is an mxn matrix, αis a scalar, thenαA is themxn matrix whose (i, j) is ij a αfor each ordered pair (i, j) .3.4 Matrix AdditionTwo matrices with the same dimensions can be added by adding their corresponding entries.★Definition : If A and B are both mxn matrices, then the sum A+B is the mxn matrix whose (i,j) entry isij ija b + for each ordered pair (i, j).An mxn zero matrix (零矩阵) is a matrix whose entries are all zero. It acts as an additive identity on the set of all mxn matrices.A+O=O+A=AThe additive of A is (-1)A since A+(-1)A=O=(-1)A+A.A-B=A+(-1)B-A=(-1)A3.5 Matrix Multiplication and Linear Systems3.5.1 MotivationsRepresent a linear system as a matrix equationWe have yet to defined the most important operation, the multiplications of two matrices. A 1x1 system can be writtena xb =A scalar can be treated as a 1x1 matrix. Our goal is to generalize the equation above so that we can represent an mxn system by a single equation.A X B=Case 1: 1xn systems 1122... n n a x a x a x b +++=If we set()12n A a a a =and12n x x X x ⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭, and define1122...n n AX a x a x a x =+++Then the equation can be written as A X b =。
线性代数英文课件：ch1_3 Cofactor Expansion
D a21 a22
a23
的余子式、代数余子式 和该元素本身无关，只
a31 a32 a33 和行列式的其他元素有关
M 23
a11 a31
a12 a32
is the complement minor of a23
A23
(1)23
a11 a31
a12 a32 is the cofactor of a23
3 1 1 2
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【 Corollary(推论)】
ai1 Aj1 ai2 Aj2 ain Ajn 0 when i j;
a1i A1 j a2i A2 j ani Anj 0 when i j;
For example,
2 1 3
A1 2 1 ,
23 4
then A11 2 A12 A13 0.
Math. Dept., Wuhan University of Technology
Proof: For determinant
a11 a12
a1n
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a21 a22
a2n
are Aj1 , Aj2 , , Ajn
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线性代数 英文讲义
Definition
A matrix is said to be in reduced row echelon form if: ⅰ. The matrix is in row echelon form. ⅱ. The first nonzero entry in each row is the only nonzero entry in its column.
n×n Systems Definition
A system is said to be in strict triangular form if in the kth equation the coefficients of the first k-1 variables are all zero and the coefficient of xk is nonzero (k=1, …,n).
1×n matrix
column vector
x1 x2 X x n
n×1 matrix
Definition
Two m×n matrices A and B are said to be equal if aij=bij for each i and j.
1 1
Matrix Multiplication and Linear Systems
Case 1 One equation in Several Unknows
If we let A (a1 a2 an ) and
Example
x1 x2 1 (a ) x1 x2 3 x 2 x 2 2 1 x1 x2 x3 x4 x5 2 (b) x1 x2 x3 2 x4 2 x5 3 x x x 2 x 3x 2 4 5 1 2 3
线性代数英文课件：ch3-1 Elementary Operations
(注：增广矩阵化为最简形时，线性方程组的解亦求出)
下列哪些矩阵是行最简形？
1 0 3 3
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I. ri rj : Interchange row i and row j.
II. kri (or k ri ) : Multiply the ith row by a nonzero
.scalar k.
III. ri krj : Add k times the jth row to the ith row (i j)
0
0
Echelon Form Matrix
Definition 4 A matrix is said to be in row echelon
form(行阶梯型矩阵), or simply an echelon matrix, if:
I. The zero rows, if any, are below all nonzero rows and
1. Elementary Operations and Gaussian Elimination Method
The Cramer’s Rule:前提：
线性代数讲义
线性代数（Linear Algebra ）引 言（Introduction ）1. 数学 数学（數學、mathematics ）在我国古代叫算（筭、祘）术，后来叫算学或数学；直到1939年6月，为了划一才确定统一用数学．“数学是研究现实世界的量的关系和空间形式的科学”，分为代数、几何等．2. 代数 代数(algebra)分为古典代数和近世代数． 古典代数(ancient algebra)基本上就是方程论，以方程的解法为中心．如： 一元一次方程 )0(≠=a b ax 的解为b a x 1-=；一元二次方程 )0(02≠=++a c bx ax 的解为)2/()4(22,1a ac b b x -±-=； 一元三、四次方程也有类似的求根公式（16世纪）；但是，一元n 次方程当n ≥5时却无一般的“求根公式”（参见数学史或近代数）；根式求解条件的探究导致群概念的引入，这最早出现在Lagrange 1770年和1771年的著作中；1799年Ruffini 给出“证明”（群论思想）；Abel 进一步给出严格的证明，开辟了近世代数方程论的道路（1824年和1826年），包括群论和方程的超越函数解法；Galois 引入代换群彻底解决了代数方程根式可解的条件，开辟了代数学的一个崭新的领域——群论．从而使代数的研究对象转向研究代数结构本身，此即近世代数．近世代数(modern algebra)又称抽象代数（abstract algebra ）包括代数数论、超复数系、线性代数、群论、环论、域论、格论、李（Lie ）群、李代数、代数几何、代数拓扑，等等． 3. 线性代数 如果保持一元一次方程中未知量的指数（一次的）不变，而增加未知量及方程的个数，即得到线性（一次）方程组．先看下面三个例子：例1 （《孙子算经》卷下第31题）“今有雉兔同笼，上有三十五头，下有九十四足．问雉、兔各几何？答曰：雉二十三，兔一十二．”解法1 设雉、兔分别为x ，y ，则由⎩⎨⎧=+=+944235y x y x 解得⎩⎨⎧==1223y x .解法2 ⎪⎪⎭⎫ ⎝⎛9435足头⎪⎪⎭⎫⎝⎛−−→−⎪⎪⎭⎫ ⎝⎛−−→−⎪⎪⎭⎫ ⎝⎛−−→−122312354735兔雉兔头半足头再作差作差半其足 . 解法 3 请兔子全“起立”后，雉兔总“足”数为70235＝⨯，从而得兔“手”数为94－70＝24，于是兔子数为24÷2＝12，雉数为35－12＝23 .注：后两种解法心算即可．例2 某厂用四种原料生产五种产品，各产品的原料成份及各原料的用量为表1所示，求每种产品的产量（千克）．表1 各产品的原料成份（％）及各原料的用量（千克）解 设A,B,C,D,E 五种产品的产量分别为X i （i ＝1，2，3，4，5），则问题归结为求解方程组 ⎪⎪⎩⎪⎪⎨⎧=++++=++++=++++=++++6001.06.01.02.01.07807.01.03.01.02.02501.02.02.06.04.01001.01.04.01.03.054321543215432154321X X X X X X X X X X X X X X X X X X X X这是一个含五个未知量、四个方程的方程组．例3 某商店经营四类商品，四个季度的销售额及利润额如表2所示．求每类商品的年平均利润率（％）． 表2 各类商品四个季度的销售额及利润额（单位：元）解 设四类商品A,B,C,D 的利润率分别为X i （i ＝1，2，3，4），则问题归结为解下面含四个未知量、四个方程的方程组 ⎪⎪⎩⎪⎪⎨⎧=+++=+++=+++=+++955005002503009075040030016085800500100200806003002002504321432143214321X X X X X X X X X X X X X X X X .现实中的很多问题，往往归结为求解含多个未知量（数）的一次方程组，称为线性方程组，其一般形式为 ⎪⎪⎩⎪⎪⎨⎧=+++=+++=+++mn mn m m n n n n b x a x a x a b x a x a x a b x a x a x a 22112222212111212111 .此类线性方程组可能有解，也可能无解；有解时可能只有一组解，也可能有多组甚至无穷多组解，如⑴⎩⎨⎧=-=+226132121x x x x 有唯一解⎩⎨⎧==03/121x x ; ⑵⎩⎨⎧=-=-226132121x x x x 有无穷多解⎩⎨⎧-==1321c x cx （其中c 为任意常数） ; ⑶⎩⎨⎧=-=-426132121x x x x 无解 .那么，如何判定一个给定的线性方程组有没有解？如果有解，究竟有多少组解（一组、多组、无穷多组）？这些解又怎样求（表示）出来？如果无解，又怎么办？因为无解的方程组如果是某一有解的实际问题的数学抽象，此时又如何（用这一线性方程组来）描述它所表示的实际问题的解（“广义解”）？这就要求我们研究解决线性方程组有解的判定条件、解法、解的结构与解的表示以及“广义解”等问题，这些都是线性代数所要解决的问题．线性代数( Linear algebra )是从线性方程组、行列式和矩阵等理论中产生出来的，是代数各分支中应用最广泛的分支．在历史上首先应归功于英国的J.J.Sylvester 、A.Cayley 、美国的Peirce 父子和L.E.Dickson 等人的工作．主要内容：行列式、矩阵、线性方程组、向量空间、相似矩阵及二次型等；主要方法：初等变换法、降阶法、分块法、标准形法、特征值法等． 下面我们将分别介绍．当然我们这里所介绍的只是线性代数中最基本的内容，还有很多内容(如矩阵论或矩阵分析等)要等到我们进一步深造时再学；而且线性代数本身也是在不断发展的．参 考 书[1] 线性代数(第三版、第四版)，同济大学数学教研室编，高等教育出版社． [2] 线性代数（居余马等编）、线性代数与解析几何（俞正光等编）、线性代数辅导（胡金徳等编），清华大学出版社． [3] 线性代数（陈龙玄等编）、线性代数（李炯生等编），中国科技大学出版社． [4] 线性代数解题方法技巧归纳，毛纲源编，华中理工大学出版社． [5] 线性代数方法导引，屠伯埙编，复旦大学出版社． [6] Linear Algebra(UTM)，By L.Smith ，Springer-V erlag ．． ． ．第一讲 行列式 ( Determinant )教学目的与要求：了解n 阶行列式的概念，掌握行列式的性质和二、三阶行列式的计算方法， 会应用行列式的性质简化n 阶行列式的计算，会用Cramer 法则解线性方程组．重点：n 阶行列式的概念、性质与计算§1 二、三阶行列式 （复习与总结）一、2阶行列例1 求下列二元一次方程组的解：(1) ⎩⎨⎧=+=+②①9442352121x x x x ；（2）⎩⎨⎧=+=+②① 22221211212111b x a x a b x a x a ……(1)（其中)021122211≠-=a a a a D ．解 (1) )1(4-⨯+⨯②①得,2346211=⇒=x x1)2(⨯+-⨯②①得1224222=⇒=x x ．(2) )(1222a a -⨯+⨯②①得121222111)(x b a a b D Dx ⇒-===D 1/D ，1121)(a a ⨯+-⨯②①得=⇒-==221121122)(x a b b a D Dx D 2/D ．为使⑴的解表示简单，Leibniz 于18世纪初引入2阶行列式的定义如下：定义 设有4个元素（数）排成的两行（row ）、两列（column ）的22211211a a a a ，称为一个2阶行列式，其值为a 11a 22－a 12a 21，即2112221122211211a a a a a a a a -=．如例1（2）中的D=22211211a a a a 称为方程组⑴的系数行列式，而2221211a b a b D =，2211112b a b a D =；（1）中的24942351,46494135,2421121======D D D ． 例２ 计算2315－=D ．解 1331252315＝）（－＝－⨯-⨯=D ． 例３设132λλ=D ，问λ为何值时，（1）D = 0，（2）D ≠0？ 解 因D =λ2－3λ＝λ（λ－3），故（1）当λ＝0或3时，D = 0；（2）当λ≠0，3时，D ≠0．例4 设1221--=k k D ，则D ≠0的充要条件是（）答：k ≠－1，3．(因D =(k －1)2－4＝（k ＋1）（k －3），故D ≠0的充要条件是k ≠－1，3) 例5 如果1222112110==a a a a D ，则下列（ ）是⎩⎨⎧=+-=+-0022221211212111b x a x a b x a x a 的解．(A)22111122221211b a b a x a b a b x ==，; (B)22111122221211b a b a x a b a b x =-=，;(C)22111122221211b a b a x a b a b x ----=----=，； (D)22111122221211b a b a x a b a b x ---=-----=，．答：（ ）(因原方程组即⎩⎨⎧-=-+-=-+22221211212111)()(b x a x a b x a x a 的系数行列式1022211211-=-=--=D a a a a D ，2221212221211a b a b a b a b D =----=，2211112211112b a b a b a b a D -=--=)二、3阶行列式例６ 求解下列三元一次方程组：（１） ⎪⎩⎪⎨⎧=++=++=++③②①333323213123232221211313212111b x a x a x a b x a x a x a b x a x a x a (2)（其中)0322311332112312213322113312312332211≠---++=a a a a a a a a a a a a a a a a a a D ；（２） ⎪⎩⎪⎨⎧-=++-=++=-③②① 232131232132131x x x x x x x x .解（1）记3332232211a a a a A =，3331232112a a a a A -=，3231222113a a a a A =, 3332131221a a a a A -=，3331131122a a a a A =,3231121123a a a a A -=,2322131231a a a a A =,2321131132a a a a A -=,2221121133a a a a A =，则： ①×A 11＋②×A 21＋③×A 31得D X 1=D 1（=b 1A 11+b 2A 21+b 3A 31）, X 1=D 1/D ， ①×A 12＋②×A 22＋③×A 32得D X 2=D 2（=b 1A 12+b 2A 22+b 3A 32）, X 2=D 2 /D ， ①×A 13＋②×A 23＋③×A 33得D X 3=D 3（=b 1A 13+b 2A 23+b 3A 33）, x 3=D 3/D ；(2) D=1+0-6-4+0-9=-18，23120A 61320A 81331A 312111＝－＝，＝－＝，＝－＝--, ，＝－－＝，＝－－－＝，＝－－＝53121A 31221A 71231A 322212--11101A 33201A 53211A 332313＝＝，＝－－＝，＝－＝-,①×A 11＋②×A 21＋③×A 31得 －18x 1＝－18 ⇒x 1＝1， ①×A 12＋②×A 22＋③×A 32得 －18x 2＝0 ⇒x 2＝0， ①×A 13＋②×A 23＋③×A 33得 －18x 3＝0 ⇒x 3＝0．定义 设有9个元素（数）排成的3行、3列的333231232221131211a a a a a a a a a 称为一个三阶行列式， 其值为322311332112312213322113312312332211a a a a a a a a a a a a a a a a a a ---++．如例6中的D 即称为方程组的系数行列式．2、3阶行列式的值（代数和）可用沙路法（或对角线法则）来记忆：211222112112221122211211a a a a a a a a a a a a -=+=,322311332112312213322113312312332211333231232221131211a a a a a a a a a a a a a a a a a a a a a a a a a a a +++++==322311332112312213322113312312332211a a a a a a a a a a a a a a a a a ---++α;或在图333231232221131211333231232221131211a a a a a a a a a a a a a a a a a a 上操作． 例7 计算 61504321-=D ． 解58051642)1(03043)1(5260105164210343152601-=⋅⋅-⋅⋅--⋅⋅-⋅⋅+-⋅⋅+⋅⋅=-+-++-+=D例8 （1）010000=-=a bb aD 的充要条件是（ ）答：022=+b a ．（因为22b a D +=）（2）0114011>=a a D 的充要条件是（），其中R a ∈．答：1012>>-a a 或．（因为12-=a D ） （3）01110212=-=k kD 的充要条件是（）（A ）k ＝2； （B ）k ＝－2； （C ）k ＝0； （D ）k ＝3．答：（B ）或（D ）．（因为)3)(2(64222-+=--=---=k k k k k k D ）例９ 计算下列行列式的值（1）749651823=D ；（2）768452913'=D解 （1）201721436032108105749651823-=---++==D ；（2）201147236010832105768452913'-=---++==D ． 三、3阶行列式的性质 (由定义易验证，对2阶也成立且验证更易)性质1 D T =D . 其中D T 为将D 的行与列互换后所得的行列式，即如果333231232221131211a a a a a a a a a D =，则332313322212312111a a a a a a a a a D T =； D T 有时也记为D ˊ，称为行列式D 的转置行列式．此性质说明在（二、三阶）行列式中行、列等位．因此凡对行（或列）成立的性质对列（或行）也成立．性质2 交换两行(或列)使行列式仅变号，即有333231232221131211333231131211232221a a a a a a a a a a a a a a a a a a -=等．对换第i 行（列）与第j 行（列）记为（r i ，r j ）（（c i ，c j ））．推论 两行（或列）相同的行列式值为0，即有0232221131211131211=a a a a a a a a a 等． 性质3 行列式中某一行（或列）的公因子可以提到行列式外面来，即有333231232221131211333231232221131211a a a a a a a a a k a a a ka ka ka a a a =等． 推论1 用数k 乘以某行列式相当于用k 乘以该行列的某一行（或列）． 以k 数乘以第i 行（列）记为)(i ic k r k ⋅⋅．推论2 某一行（或列）全为0的行列式的值为0．推论3 有两行（或列）成比例的行列式的值为0．如0333231131211131211333231131211131211==a a a a a a a a a k a a a ka ka ka a a a 性质4 若行列式的某一行（或列）的每个元素都是两个元素之和，则此行列式可按此行（或列）分拆成两个行列式之和．如=+++=333231232322222121131211a a a cbc b c b a a a D D 1＋D 2，其中3332312322211312111a a a b b b a a a D =，3332312322211312112a a a c c c a a a D =． 性质5 将某一行（或列）各元素的同一数倍加于另一行（或列）相应的元素上去，不改变行列式的值，即有333231232221131211333231132312221121131211a a a a a a a a a a a a la a la a la a a a a =+++等． 将第j 行（列）的l 倍加到第i 行（列）记为 r i ＋⋅l r j （ c i ＋⋅l c j ）．注：性质2、3和5中的变换：对换两行（或列）、以非零常数乘某行（或列）和把某行（或列）的常数倍加到另一行（或列）上去，分别称为第一、第二和第三类初等行（或列）变换（详见第二讲§5）．性质6（按行（或列）展开定理） （1）∑====31333231232221131211)3,2,1(j ij ij i A a a a a a a a a a a D ，即333332323131232322222121131312121111A a A a A a A a A a A a A a A a A a D ++=++=++=；（2）∑==31i ij ijA aD （j=1,2,3）, 即313121211111A a A a A a D ++=333323231313323222221212A a A a A a A a A a A a ++=++=（其中A ij 如例6所示：ij ji ij M A +-=)1(，M ij 是将Ｄ中a ij 所在的第i 行和第j 列全划掉余下的二阶行列式，称为a ij 在D 中的余子式，而A ij 称为a ij 在D 中的代数余子式．） 例10 计算下列行列式的值（1）151413---=kk D ；（2）12121-=k k kD ． 解（1）)3)(1(1430114004315140132321++=-+-=-+++---=k k k k k k r r r r k k D ；；（2）)2(222020021121211312k kk k kkr r r r kk kD --=--=-----=；．性质7（代数余子式的性质） （1）D A a ik j kj ij δ=∑=31（其中⎩⎨⎧≠=ki 0ki 1，＝，δik 为Kronecker 记号．当i ＝k 时即为性质6（1）；当i ≠k ，如i ＝1，k ＝2时，0A 231322122111=+A a A a a ＋等）．（2）D A ajl i il ijδ=∑=31（当j ＝l 时即为性质6（2）；当j ≠l , 如j ＝1，l ＝2时， 0A 323122211211=+A a A a a ＋等）．例11 求132311201--=D 的值，并验证性质7．解 D 的23120A 61320A 81331A 312111＝－＝，＝－＝，＝－＝-- ，＝－－＝，＝－－－＝，＝－－＝53121A 31221A 71231A 322212--,11101A 33201A 53211A 332313＝＝，＝－－＝，＝－＝-(1) 按第1列展开得=⋅-⋅+⋅=312111211A A A D 1×(-8)+1×(-6)-2×2=-18；(2) 023)6(1)8(0310312111=⋅+-⋅+-⋅=⋅+⋅+⋅A A A ；其余类似．四、Cramer 法则1．一般情形 由例1和例6即得定理（Cramer 法则） （1）二元一次线性方程组 ⎩⎨⎧=+=+②① 22221211212111b x a x a b x a x a …（1）当其系数行列式D=22211211a a a a ≠0时有唯一解D D x j j =（j ＝1，2）； （2）三元一次线性方程组⎪⎩⎪⎨⎧=++=++=++③②①333323213123232221211313212111b x a x a x a b x a x a x a b x a x a x a …（2）当其系数行列式D=333231232221131211a a a a a a a a a ≠0时有唯一解 D D x j j =（j ＝1，2，3）． 例12(例6(2)的解法2 ) 18132311201＝－--=D ，D 1＝D ＝－18， 01223112112=---=D ，⎪⎩⎪⎨⎧⇒--0x 0x 1x 023211111D 3322113＝＝＝＝＝＝＝＝D D D D D D ．注：两种解法本质是一样的，只不过解法2是直接用Cramer 法则的结果（公式），而原解法是把消元（或Cramer 法则的证明）过程再写一遍．2．齐次情形推论 奇次线性方程组 ⎪⎩⎪⎨⎧=++=++=++000333232131323222121313212111x a x a x a x a x a x a x a x a x a （2ˊ）当其系数行列式D ≠0时只有零解（ｘ1＝ｘ2＝ｘ3＝0）．以后将证明此推论的逆也成立，于是有命题（1）奇次线性方程组（2ˊ）只有零解⇔D ≠0；（2）奇次线性方程组（2ˊ）有非零解⇔D ＝0．例13 λ取何值时，奇次线性方程组⎪⎩⎪⎨⎧==-+=++00023321321x x x x x x x λλλ，有非零解？解 因为)1(0011212-=-=λλλλλD ，故当λ＝0或±1时，该方程组有非零解．例14 如果方程组⎪⎩⎪⎨⎧=--=+=-+05403z y kx o z y z ky x 有非零解，则（）．（A ） k ＝0；（B ）k ＝1：（C ）k ＝－1；（D ）k ＝－3．答：（C ，D ）．(由例10（1）)3)(1(1514013++=---=k k kk D 即得) ．例15 当（）时，奇次线性方程组⎪⎩⎪⎨⎧=+-=++=+02020z y kx z ky x z kx 仅有零解；（A ） k ＝0；（B ）k ＝－1；（C ）k ＝2；（D ）k ＝－2．答：（A ，B ，D ）．(由例10（2）)2(2121210k kk kD --=-=即得) ．§2 全排列及其逆序列问题：行列式可否归纳定义 212112221111222112112)1()1(a a a a a a a a D ⋅-⋅+⋅-⋅==++，当n ≥2时，n n nn ijn A a A a A a a D 1112121111+++==⨯ ，其中j j j M A 111)1(+-=，M 1j 为a 1j 在D n 中的余子式（n －1阶行列式）？一、全排列例1 用1、2、3三个数字可以组成多少个没有重复数字的三位数？请写出．解 共6个，分别为123，132，213，231，312，321．把n 个不同的元素（不妨设为1，2，…，n ）排成一列，叫做这n 个元素的一个（n 级）全排列，n 个不同元素的全排列的种数为P n ＝n!，如P 3＝3!＝6，P 4＝4!＝24，P 5＝5!=120等．记S n 为1，2，…，n 的所有n 级排列所组成的集合，即S n ＝{（j 1j 2…j n ）| (j 1j 2…j n )为n 级排列}，则|S n |= n!．二、逆序与逆序数1. 标准排列：对n 个不同的元素，先规定一个标准次序（如对1，2，…，n ，规定从小到大的次序为标准次序），从而得到一个标准排列．对1，2，…，n ，今后规定其标准排列为自然排列1 2 … (n －1) n ．2.逆序与逆序数 在一个n 级排列中，当两个元素a 和b 的先后次序与标准顺序不同时，则说a 和b 形成一个逆序；一个排列中所有逆序的总数叫该排列的逆序数．排列p 1p 2…p n 的逆序数记为 t （p 1p 2…p n ）．逆序数为奇（或偶）数的排列称为奇（或偶）排列． 例2 （1）2个2级排列12和21，一个为奇排列（21），一个为偶排列（12）．（2）3级排列的逆序数表（6个3级排列中奇、偶排列各3个）三、逆序数的求法不妨设n 个元素为1，2，…，n ，其标准排列为自然排列1 2 … n ，设p 1p 2…p n 为1，2，…，n 的一个排列，记t i =t(p i )为排列p 1p 2…p n 中p i 左（前）面的比p i 大的元素的个数，s i ＝s （p i ）为排列p 1p 2…p n 中p i 右（后）面的比p i 小的元素的个数，简记t(p 1p 2…p n )为t ，则(1)t(1)t(2) t(n)t(n) t(2)t(1)t t t t n 21+++=+++=+++= ；(2)s(1)s(2) s(n)s(n) s(2)s(1)s s s n 21+++=+++=+++= t ． 注：显然0(1)t(n)t 1====s s n ， t(1 2 … n)=0．例３ 求（1）t=t(32514)；（2）t(7632451)；（3）t(2 3 … n 1)；（4）t(n (n-1) … 2 1) ． 解 （1）因t 1=t(3)=0，t 2=t(2)=1，t 3=t(5)=0，t 4=t(1)=3，t 5=t(4)=1⇒ t =5（奇）， 或因s 1=s(3)=2，s 2=s(2)=1，s 3=s(5)=2，s 4=s(1)=0，s 5=s(4)=0⇒ t =5． （2）t(7632451)=0+1+2+3+2+2+6=16，或=6+5+2+1+1+1+0=16（偶）． （3）t(2 3 … n 1)=0+0+…+0+(n-1)= n-1，或=1+1+…+1+0= n-1．（4）t(n … 2 1)=0+1+…+(n-2)+(n-1)=2)1(-n n ，或=(n-1)+(n-2)+…+1+0=2)1(-n n ．如62341) 2 3 t(4=⋅=，102451) 2 3 4 t(5=⋅=，152561) 2 3 4 5 t(6=⋅=，… 再看6个3级排列的逆序数：t(123)=0，t(132)=0+0+1=1，或=0+1+0=1；t(213)=0+1+0=1，或=1+0+0=1；t(231)=0+0+2=2，或=1+1+0=2；t(312)=0+1+1=2，或=2+0+0=2；t(321)=0+1+2=3，或=2+1+0=3．四、对换及其性质１．对换：在一个排列中，互换某两个元素（如i ，j ）的位置，而其余的元素不动，叫做对该排列的一次对换，记为（i ，j ）；互换相邻两个元素的对换叫做相邻对换．例４ （321）0)123(,3)321(),123(3,1(==−−→−t t ）；（7632451）−−→−)1,6(（7132456），t(7632451)=16（偶）， t(7132456)=0+1+1+2+1+1+1=7（奇）．２．性质性质1 一次对换改变排列的奇偶性．证明（1）相邻对换改变排列的奇偶性：设（…a b …）−−→−),(b a （…b a …）因对换（a ，b ）只改变了a 和b 之间的逆序：当a<b 时，经对换后逆序数增加1；当a>b 时，经对换后逆序数减少1．而a 或b 与其他元素，以及其他元素之间的逆序数经对换后都没有改变，故相邻对换改变排列的奇偶性．（2）任一对换可由奇数次相邻对换而得到，从而改变奇偶性：（…ac 1c 2…c s b …）−−→−),(1c a(…c 1ac 2…c s b …) −−→−−−→−),(),(2sc a c a （…c 1c 2…c s ab …）−−→−),(b a （…c 1c 2…c s ba …）−−→−),(b c s (…c 1c 2…bc s a …)−−→−−→−),(2b c ( …c 1bc 2…c s a …)−−→−),(1b c (bc 1c 2…c s a …)，共2s+1次．例４（1）对换（7632451）−−→−)1,6(（7132456）可由9次相邻对换得到，相应的逆序变化为：（16=）t(7632451)= t(7362451)+1= t(7326451)+2= t(7324651)+3= t(7324561)+４ = t(7324516)+5=( t(7324156)+6= t(7321456)+7= t(7312456)+8= ) t(7132456)+9 (=7+9)． （2）(7=) t(7132456)= t(1324567)+6= t(1234567)+7 (=0+7)．（3）(16=) t(7632451)= t(6324517)+6= t(3245167)+11= t(3241567)+12= t(3214567)+13=t(2134567)+15=t(1234567)+16(=0+16)．性质2 （1）任一排列（p 1p 2…p n ）总可经有限次（相邻）对换成标准排列，且所作对换的次数k 与该排列有相同的奇偶性，即k 与t (p 1p 2…p n )奇偶性相同；（2）任一排列（p 1p 2…p n ）都可由标准排列1 2 … n 经有限次（相邻）对换而得到，且所作对换的次数k 与该排列有相同的奇偶性，即k 与t (p 1p 2…p n )奇偶性相同．（3）S n 中的任意两个n 级排列均可经有限次（相邻）对换而互相得到；且若这两个排列的奇偶性相（或不）同，则所作对换的次数为偶（或奇）数．证（1）对排列的阶n 归纳．当n=1时显然成立．假设结论对n －1已经成立，则对n ： ①若排列为p 1 … p n-1 n ，由归纳假设n －1级排列p 1 … p n-1可经有限次对换成为标准排列1 2 … （n －1），且所作对换的次数与t(p 1…p n-1)有相同的奇偶性，从而p 1…p n-1 n 经上述对换即成为标准排列1 2 …(n －1) n ，且所作对换的次数的次数与t(p 1…p n-1 n)＝t(p 1…p n-1)有相同的奇偶性．②若排列为（p 1…p i-1 n p i+1…p n ），则可经n －i 次相邻对换成为（p 1…p i-1p i+1…p n n ），且t （p 1…p i-1 n p i+1…p n ）=t （p 1…p i-1p i+1…p n n ）＋（n －i ），而由①得p 1…p i-1p i+1…p n n 可经有限次对换成为1 2 …(n －1) n ，且所作对换的次数m 与t （p 1…p i-1p i+1…p n n ）有相同的奇偶性，于是p 1…p i-1 n p i+1…p n 可经m ＋n －i 次对换成为1 2 … n ，且所作对换的次数k =m+n-i 的奇偶性与t （p 1…p i-1 n p i+1…p n ）即t （p 1…p i-1p i+1…p n n ）+n －i 的奇偶性相同．图示如下：（p 1…p i-1 n p i+1…p n ）−−→−-次i n （p 1…p i-1p i+1…p n n ）−−→−次m （1 2 …(n －1) n ）； ③所作对换次数与原排列有相同的奇偶性还可如下证明：设排列p 1…p n 经k 次对换成为标准排列，则t(p 1…p n )经k 次改变奇偶性后成为0 (=t(1 2 … n))，从而k 与t(p 1…p n )奇偶性相同（对k 为奇、偶数分别说明）．（2）将（1）中的变（对）换全倒过来便得．（3）由（1）和（2）：−→−次k n p p p )(21 （1 2 … n ）)21h n q q q （次−→−即得．性质3 n ！个n 级排列中奇偶排列各为 ）2(2!≥n n ．证 因映射ϕ：{n 级奇排列}−−→−），（21{n 级偶排列}为一一对应，即得． 如 {(21)}−−→−），（21{(12)} ； {(132), (213), (321) }−−→−），（21{(231), (123), (312)} .§3 n 阶行列式的概念一、 二、三阶行列式的结构规律１. 二、三阶行列式定义式的结构（1）2112221122211211a a a a a a a a -=中的两项可统一表示为212121)()1(j j j j t a a -，其中（j 1j 2）取遍所有（2个）2级排列（12），（21）．（2）33⨯ija ＝322311332112312213322113312312332211a a a a a a a a a a a a a a a a a ---++α中的6项可统一表示为321321321)()1(j j j j j j t a a a -，（j 1j 2j 3）取遍所有（6个）3级排列（123），（231），（312），（321），（213），（132）．２. 二、三阶行列式的共同规律设n ＝2或3，则：(1) n 阶行列式为由n 2个数得到一个数的函数；(2) n 阶行列式为n ！项的代数和，每项为n 个元素的乘积，而这n 个元素是取自n 阶行列式中的不同的行、不同的列；(3) n 阶行列式中每项正负号的确定：当项中各乘积因子的第一个（行）下标为标准排列时，其第二个（列）下标为奇（偶）排列的项带负（正）号． 3. 二、三阶行列式的简单统一表达式（1）∑∈-=221221)(211)(22211211)1(s j j j j j j t a a a a a a ，其中)}1,2(),2,1{(2=S ；（2）∑∈-=332132321)(3211)(333231232221131211)1(s j j j j j j j j j t a a a a a a a a a a a a ，S 3＝{（j 1j 2j 3）| (j 1j 2j 3)为3级排列}={（123），（231），（312），（321），（213），（132）}．二、n 阶行列式的定义1．定义 设有n 2个元素排成的n 行、n 列的nnn n nn a a a a a a a a a 212222111211称为一个n 阶行列式，其值为∑∈-nn n n s j j j nj j j j j j t a a a )(211)(21221)1( ．上述n 阶行列式可简记为nn ij a ⨯或det n (ij a )． 注：⑴当n ＝2，3时，与前面定义一致；当n ＝1时，1111a a =（注意别与绝对值混淆）． ⑵当n ≥4时，“沙路法”不再成立（或不再那样简单），见例1(4)．例１ （1）n 21n21λλλλ０λ０λ=（（主）对角线（形）行列式），n nn a a a a ＝⨯0，11101＝， |0|n ×n = 0； （2）nn nnn n a a a a a a a a a 221121222111＝（下三角形行列式）；（3）nn nn nna a a a a a a a a2211222112110=上三角形行列式）(；（4）)1(0112211111111212)1(112121n n nn n n n n n nn nn n n n na a a a a a a a a a a a a a a------=-=，n n n λλλλλλ212)1(21)1(0--=ｎ（次对角形行列式）；如,abcd 0dcb a 0= ,abcde 0ed cb a= ;abcdef 0fed c b a 0-=（5）abcd abcd abcd d c b a t -=-=-=3)3142()1()1(000000000000； （6）111111)1(11001001011010)4123(-=-=⋅⋅⋅-=３）（t （因第3行和第1列均只有一个非零元素，因此非零项必取含21a 32a 的，从而另两个乘积因子11j a 和44j a 只能分别取14a 和43a 才能使该项不为0，于是得结果）； （7）∑∑∈∈-⋅=-=34324324324432432432)(432)(11S )j j (1j 43211)1(44434241443332312423222111)1()1(000S j j j j j j j j j t j j j j j j t a a a a a a a a a a a a a a a a a a a a a })243,324,432,423,342,234{}3234{(344434234333234232211）（）（）（）（）（）（＝级排列的=⋅=S a a a a a a a a a a ；类似有nnn n nnn n na a a a a a a a a a a a2222112122221110⋅=，特别地，00002122221=nnn n na a a a a a，一般地，nnnr nr r r rrr r nnr n nrn n r r r r r r rrr r a a a a a a a a a a a a a a a a a a a a11111111111111111111000++++++++++⋅=， 简记为C A CB O A ⋅=;（8）当n ≥2时，n n n n n nn n a a a a a a a a a a 12212)2)(1(1221)1(0000000000000-------=．（9）当n ≥2时，02!2!)1(1111111111)()(2121=-=-==∑⨯n n n n j j j j j j t nn ． （10）nn ijj n nj j j j j j j j t nn ji j i a b a b a b a b ba n n n ⨯---⨯-=-=≠∑)())(()1()0(2211212211)( ．2．等价定义定理1 n i i i i i i i i i t nn ijn n n a a a a D 21)()(212121)1(∑-==⨯（记为D 1）．证 ⎢⎢⎢⎣⎡−−−−−−−−→−⎥⎥⎥⎦⎤)n 12)12()(2121i i j j 212121i j j i 21n nj j j n n i i i j j j a a a n i i i a a a nn 列标（）行标（列标行标）＝，＝经若干次对换（ 因n n n nj j j j j j j j j t nn ija a a a 21212121)()()1(D ∑-=⨯＝由对换的性质2知对D 中任一项n n nj j j j j j t a a a 212121)()1(-总有且仅有D 1中的某一项n i i i j j j t n n a a a 21)(2121)1(-与之对应并相等；反之，对D 1中任一项n i i i i i i t n n a a a 21)(2121)1(-，也总有且仅有D 中的某一项n n nj j j j j j t a a a 212121)()1(-与之对应并相等，如Ｄ4中))1(()1())1(()1(42342113334134221)2413(42342113342342113)3142(a a a a a a a a a a a a a a a a t t -=-=-=-；于是D 与D 1中的项可以一一对应并相等，从而D ＝D 1． 定理2 n n j i j i j i J t I t nn ija a a a 2211)()()1(∑+⨯-=，其中t(I)=t(i 1i 2…i n )，t(J)=t(j 1j 2…j n )，∑为对所有n 级排列(i 1i 2…i n )求和（此时(j 1j 2…j n )为某一固定的n 级排列），或为对所有n 级排列(j 1j 2…j n )求和（此时(i 1i 2…i n )为某一固定的n 级排列）．证 用对换的性质2（3），与定理1类似证明即可．再看例1（3），a b c da b c d a c b d a b c d b d a c d c b a t t t -=-=-=-=-=++41)3412()1324()2413()1()1(,)1(000000000000又．注：此例中i 1＝2，i 4＝4，i 3＝1，i 4＝3；j 1＝3，j 2＝1，j 3＝4，j 4＝2；j i1＝j 2＝1，j i2＝j 4＝2，j i3＝j 1＝3，j i4＝j 3＝4；i j1＝i 3＝1，i j2＝i 1＝2，i j3＝i 4＝3，i j4＝i 2＝4．§4 行列式的性质一、性质设nnn n n na a a a a a a a a D212222111211=，记D T（或D ˊ）nnn nn n a a a a a a a a a212221212111=，称为D 的转置（行列式），由§3定理1立即得：性质1 D T＝D , 即任一行列式与其转置的值相等．此性质说明：行列的行与列具有同等的地位，凡对行（或列）成立的性质对列（或行）也成立．性质2 互换行列式的两行（或两列），行列式仅变号．证 设k ia a a a D in i kn k111=，欲证D 1＝－D ，只需证D 1和D 的定义式中的一般项互为相反数即可．事实上，D 1中的一般项为n k i 1n k i 1nj ij kj j 1)j j j j (t a a a a )1( -n i k 1n i k 1nj kj j i j 1)j j j j (t a a a a )1( --=恰为D 中一般项的相反数；故得证．推论 两行（或列）完全相同得行列式值为零．性质3 行列式某一行（或列）的公因子可以提到行列式外面来，kD a a ka ka a a nnn in i n=11111． 证明与性质2的证明类似，考虑一般项即可．推论1 行列式的某一行（或列）中所有元素都乘同一数k ，等于用k 乘此行列式． 推论2 某行（或列）全为零的行列式的值为零． 推论3 两行（或列）成比例的行列式的值为零．性质4 若行列式中某一行（或列）的元素都是两项之和，则该行列式可按此行（或列）分拆成两个行列式之和，即nnn in i nnnn in i n nnn in in i i n a a c c a a a a b b a a a a c b c b a a1111111111111111+=++． 证明与性质2的证明类似，考虑一般项即可．性质5 将行列式某一（如第j ）行（或列）每个元素的常数l 倍加到另一（如第i ）行（或列）相应的元素上去，其值不去，即nn ijnnn jn in j i na D a a la a la a a a ⨯==++(111111)．证 由性质4，左边的行列式可分拆成两个行列式之和，一个为D ，而另一个为0111111=nnn jn j jn j na a a a la la a a（因其第i 行与第j 行成比例）；从而得证．二、行列式的计算—化为三角形行列式定理1 任何一个行列式均可利用性质2和5化为上（或下）三角形行列式，从而计算其值．证 （1）若a ij ＝0（i ，j ＝1，2，…，n ），则00==⨯n n D ；（2）若0≠∃ij a ，则可用性质2（先第1行与第i 行互换，再第1列与第j 列互换）将a ij 调到左上角；（3）若011≠a ，则可用性质5将第1列（或行）的其余n －1个元素化为零（“打洞”）； （4）对右下角的n －1阶行列式重复（1）～（3）的步骤，如此下去（归纳），即可将D 化为上（或下）三角形行列式．以下以（r i ，r j ）表示互换i ，j 行；r i ＋hr j 表示将第j 行的h 倍加到i 行． 例1（1）4130211021102011)r 2r (),r r (0112012121102011)r ,r (0112012120112110141321-------+-----------;4)2()2()1(12000420021102011r r 2200420021102011)r 3r (),r r (342423=-⋅-⋅-⋅-=-------------++（2）)r ,r (72160112064802131)r 5r (),r r (3315112043512131)c ,c (335111024315211332141221------+--------------)r 4r (108003200112021315)r ,r (1510001080011202131)r 8r (),r 4r (7216064801120213134432423-----⋅-----+-----402221520003200112021315=⋅⋅⋅⋅=---⋅．（3）48222162002000020111164,3,2i ),r r (31111311113111116)r r r (r 31111311113111131i 4321=⋅⋅⋅⋅=⋅=-⋅+++ ; （4）xaa aa a x a a a a a x a a a a a x a a n x n i r r xaa aaa x a a a a a x a a a a a x a a a a a x i11111])1([,,2),(1⋅-+=+11)(])1([000000000000011111])1([,,2),(--⋅-+=----⋅-+=-n i a x a n x ax ax a x ax a n x n i ar r;（5）cb a b a ac b a b a a c b a b a a dc b a i r rd c b a c b a b a a d c b a c b a b a a d c b a c b a b a a d c b a i i +++++++++=-+++++++++++++++++++3630232001,2,3),(361036323423214341030020002,3),(a aa aa r rb a a b a a a a i r r i i =*-++*=-+．例2 证明奇数（n ）阶反对称行列式（a ji ＝－a ij ）的值为零，即000021212112=---n nnna a a a a a ．证 0)1(=⇒-=⋅-==D D D D D n T ．例3 解方程 (a 1≠0)113211232113221132111321=-+-+-+-+-------xa a a a a a a xa a a a a a a x a a a a a a a xa a a a a a a a n n n n n n n n n n n n解 将左边行列式的第1行的相反数分别加到第2～n 行，得左边x)-x)(-(x)-x)(-(00000000000001-n 2-n 21112211321a a a a a xa xa x a x a a a a a a n n n n=----=---故原方程的解为)1,,2,1(-==n i a x i i ，共n －1个解．三、按行、列展开定理1．代数余子式 设nn ji a D ⨯=，把D 中元素a ij 所在的第i 行和第j 行划去后，余下的n －1阶行列式叫做a ij 在D 中的余子式，记作M ij ，记A ij ＝（－1）i ＋j M ij ，叫做a ij 在D 中的代数余子式．例４（1）213132321----=D 的52113)1(1111=--⋅-=+A12312)1(2112=---=+A ,71332)1(3113-=---=+A ；（2）5021011321014321---=D的19521013201)1(3113=---=+A ,521013421)1(3223---=+A =- 63，18521201421)1(3333=--=+A ．10013201421)1(3443-=--=+A ；2．按行、列展开定理引理 若n 阶行列式nn ij a D ⨯=的元素a ij 所在第i 行（或第j 列）的其他所有元素全为零，则ij ij A a D =．证 （1）当i ＝j ＝1，即D 的第1行（或第1列）除a 11外所有元素全为零，则由§3例1（7）知1111A a D ⋅=；（2）一般地，设nnnjn ijnja a a a a a a D1111100=，将D 的第i 行依次与第i －1，i －2，…，2，1行对换，再将第j 列依次与第j －1，j －2，…，2，1列对换，使a ij 调到左上角，所得的新行列式D D D j i j i ⋅-=⋅-=+-+)1()1(21，而a ij 在D 1中的余子式即为a ij 在D 中的余子式M ij ，由（1）ij ij ji ij ij A a D D M a D =-=⇒=+11)1(．定理2 n 阶行列式nn ija D ⨯=的值等于其任一行（或列）的每一个元素分别与其相应的代数余子式的乘积之和，即),,2,1(111n i A a A a A aD in in i i nj ij ij=++==∑=或∑==++==ni nj nj j j j ij n j A a A a Ai a D 111),,2,1( ．证 （1）nnn n i nnnn n in i i n a a a a a a a a a a a a a a a a D2111121121211121100000000=+++++++++=),,2,1(00002211211121121211211n i A a A a A a a a a a a a a a a a a a a a in in i i i i nnn n in nnnn n i n=++++++引理 （2）由行列式的性质1立即得对列的等式也成立．例4 （3）对（1）中的18)7(312513)2(1131211-=-⋅+⋅-⋅=⋅+⋅-+⋅=A A A D ；对（2），24)10(018)1()63(1193-=-⋅+⋅-+-⋅+⋅=D ． 定理3 设nn ija D ⨯=，则（1）D A a A a A aik kn in k i nj kj ij⋅=++=∑=δ 111；（2）∑=⋅=++=ni jr nr nj r j ir ijD A a A a A a111δ证 （1）由定理1知当i ＝k 时成立．当i ≠k 时，将nn k ia a a a a a a a a a a a nn n n in i i in i i n⨯=212121112110按第k 行展开即得∑==nj kj ijA a10，即∑=≠⋅=nj kj ij k i D A a 1)(0；故得证．由行列式的性质1立即得对列的结论（2）也成立．定理2、3表明，行列式D 的任一行（或列）的每一个元素与其相应的代数余子式的乘积之和等于D 的值，而D 的任一行（或列）的每一个元素与另外一行（或列）的每一个元素的代数余子式的乘积之和等于零．例４（4）对（1）中的D 有 0)7()1(1352)1(32131211=-⋅-+⋅+⋅-=⋅-+⋅+⋅-A A A ， 0)7(211532)1(3131211=-⋅+⋅-⋅=⋅+⋅-+⋅A A A ；对（2）中的D 有0)10(1183)63(1191131143332313=-⋅+⋅+-⋅+⋅=⋅+⋅+⋅+⋅A A A A ， 0)10(2181)63(01922)1(024*******=-⋅+⋅--⋅+⋅=⋅+⋅-+⋅+⋅A A A A ， 0)10(5180)63(2194)5(024********=-⋅-⋅+-⋅+⋅=⋅-+⋅+⋅+⋅A A A A ．3. 行列式的归纳定义 11111a a D ==,21122211212112221111222112112)1()1(a a a a a a a a a a a a D -=⋅-⋅+-⋅==++，当n ≥2时，n n nn ijn A a A a A a a D 1112121111+++==⨯ ，其中j j j M A 111)1(+-=，M 1j 为a 1j 在D n 中的余子式（n －1阶行列式）．可以证明如上定义的n 阶行列式与前面的定义n 阶行列式是完全一样的．4．行列式的简化计算 首先利用性质将某行（或列）化为仅有一个元素可能非零，再按该行（或列）展开，降为n －1阶行列式，如此下去，直到化为二阶或一阶，即可计算其值． 例5（1）527211417)1()1(5207011321014107)2(),2(5021011321014321233431---⋅-=----++---+r r r r 241861926)1(110921126)2(),(222321-=--=-⋅-⋅=-+-+r r r r ．（2）)4)(1(22)1(202001120020001100112002000110011212--=-=-=--k k kk k k k k kk k r r k k k ． （3）0551*******3550100131111115)(),2(33511102431521133431----=----+-------c c c c 40552605502611512=---=----r r ．例6（1）dd c dcb a b a a dcd c dc b a baba D n 000012⋅=行展开按第)1(2)1(21)12()1(2)()1(000--+---=--=⋅-n n n n D bc ad D bc adD cd c dc b a bab（递推公式）)()()()(221)2(22bc ad dc ba D bc ad D bc ad D bc ad n n n -==-=-==-=-- ． （2）n ≥2，ba b ba bb ab a ba b a a a b b a ba ba D n n 0)1(010001+-⋅+⋅=列展开按第n n b a )(--=．（3）++---+---=----xa a a a xxxx xa a a a a xxxx D n n n n n n 122112211111111列展开按第n n n n n nn a a xD x a xD x xa +++=--------+)((1111)1(1211递推公式）＝).1,(21212211111a x a x xa a x D x a D a x a x a x n n n n ++=+-=+=++++==--例7（1）计算V andermonde 行列式)2(≥n ：Ｄn =),,,(21n x x x V =),(111112112222121j i x x x x x x x x x x x j i n nn n n n ≠≠---； 解 D 2 =122121211),(x x x x x x V -==，将D n 中依次第i 行减去第i －1行的x n 倍。
线性代数英文课件2.2
Definition: • A collection of a finite number of linear equations involving the same variables is called a system of linear equations, often abbreviated as SLE. • A system of m linear equations in n variables is called an m × n SLE (pronounced “m by n”). • An n-vector (x1 , x2 , ..., xn ) is called a solution to a particular SLE with n variables if it satisfies all of the equations in the system.
Math 1229A/B
Unit 5: Systems of Linear Equations
(text reference: Section 2.2)
Байду номын сангаас
c V. Olds 2010
Unit 5
61
5
Systems of Linear Equations
You know what the standard form of an equation of a line in ℜ2 looks like. It has the form ax + by = c, for some constants a, b and c. Any equation in 2 variables in which each variable is only multiplied by a constant is the standard form equation of a line in ℜ2 . The variables are usually called x and y , but they could be called x1 and x2 , or m and n, or anything. Of course, we can have other kinds of equations with 2 variables, and you’ve probably seen some of those before. For instance, x2 + y 2 = 1 is the equation of the circle with radius 1. More complicated curves in 2-space have equations like x2 + xy + y 2 = 4 or 5x4 y − 3x2 y 2 + 2xy 5 = 0 or √ √ y x − x y + y = 3. And then there are equations like x + x − y22 = 6, and 2x+y = 4 and x + sin y = 1. But none of those are linear equations. They’re not lines. They’re curvy things. You also know that in ℜ3 , an equation like x + y + z = 1 is not a line. It’s a plane. But ... what is a plane? It’s just a whole bunch of lines side-by-side. Well, okay, we wouldn’t really think of a plane that way, but a plane does have some characteristics which make it similar to a line in some ways. It’s flat. No curvy bits. And the same goes for a hyperplane in ℜm . Any plane or hyperplane, in standard form, has an equation which is just the sum of a bunch of constant multiples of variables, set equal to some constant. There are never any more complicated things done to the variables, like squaring, or taking the square root, or multiplying two variables together, or dividing by a variable. Just a constant times a variable, plus a constant times another variable, plus ... and equal to some constant. The most interesting things that happen are that sometimes a constant is negative, so that we’re actually subtracting, or sometimes a constant is 0, so that the variable isn’t even there. Equations like this are called linear equations, because the relationship between the variables is always like the relationship between the variables in an equation of a line in ℜ2 . We’re going to be working with linear equations a lot in this course. In fact, the textbook is called Elementary Linear Algebra, because the kind of algebra we’re studying all relates to these linear equations. So we need a careful definition of what these are. But it’s not any more complicated than what we’ve already said. Definition: A linear equation in the variables x1 , x2 , ... xn is an equation which can be put into the form a1 x1 + a2 x2 + ...an xn = b for some constants a1 , a2 , ..., an and b, where not all of the ai values are 0. Note: That last bit just means that the equation 0 = 0 isn’t considered to be a linear equation. There has to be at least one variable actually appearing in the equation. Also, as you already know, the variables don’t necessarily have to have the names shown in the definition. They could be x, y, z, w, r and t, or they could be subscripted y ’s or s’s instead of x’s. The names of the variables don’t matter. Any equation which says, or can be rearranged to say, that the sum of scalar multiples of variables is equal to a constant is a linear equation.
线性代数英文ppt5
Ch6_12
Composition of Matrix Transformations
Let T1: Rn → Rm and T2: Rm → Rs be matrix transformations defined by T1(x) = A1x and T2(x) = A2x. We shall now see that T2。 T1 is defined by the product matrix A2A1. T2。T1(x) = T2(T1(x)) = T2(A1x) = A2A1x
- 2 8 and T 4 17 1
e.g.,
1 1 T 2 11 3
Ch6_7
Theorem 6.1
Let A be an mn matrix. Let x be an element of Rn, interpreted as a column matrix. The transformation T: RnRm, defined by T(x)=Ax, is linear. Such a linear transformation is called a matrix transformation.
Example 4
Let D be the operation of taking the derivation. (D is he same as d d . It is a more appropriate notation in this context than .) dx dx D can be interpreted as a mapping of Pn into itself.
线性代数英文课件3.1
Unit 7: Matrix Operations
(text reference: Section 3.1)
c V. Olds 2010
Unit 7
91
7
Matrix Operations
At the beginning of Unit 6, we defined the mathematical construct called a matrix. Next we learn more about matrices, especially how to do matrix arithmetic. First, we should review the definition of a matrix. And define one more term, that we didn’t need in what we have done so far. Definition: A matrix is a rectangular array of numbers: a11 a12 a13 · · · a1n a21 a22 a23 · · · a2n . . . . . . . . . . . . am1 am2 am3 · · · amn
has 5 rows and 2 columns, so it is a 5 × 2 matrix. The (1, 2)-entry is 2
92
ห้องสมุดไป่ตู้
Unit 7
0 (c) We have the matrix 1 , which has 3 rows and only 1 column, so this is a 3 × 1 column vector. 2 Since it doesn’t have any column 2, there is no (1, 2)-entry. The (2, 1)-entry is 1. 3 2 , has only 1 row, with 4 columns, so it is a 1 × 4 row vector. The 7 (1, 2)-entry is 0, and of course there is no (2, 1)-entry. −3 0
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(f )( x) f (x)
The Vector Space Pn
Pn denote the set of all polynomials of degree less than n.
( p q)( x) p(x) q(x)
that x+(-x)=0. A5. α(x+y)= αx+ αy for each scalar α and any x and y in V. A6. (α+β)x=αx+βx for any scalars α and β and any x ∈V. A7. (αβ)x=α(βx) for any scalars α and β and any x ∈V. A8. 1·x=x for all x ∈V.
defined by
x1
x
x2
xn
x1 y1
and
x
y
x2
y2Leabharlann xn yn for any x, y Rn and any scalar 。
The Vector Space Rm×n
Rm×n denote the set of all m×n matrices with real entries.
Mathematical systems of this form are called vector spaces or linear spaces.
1 Definition and Examples
Euclidean Vector Spaces Rn
In general, scalar multiplication and addition in Rn are
Vector Space Axioms
Definition
Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that, with each pair of elements x and y in V, we can associate a unique elements x+y that is also in V, and with each element x in V and each scalar , we can associate a unique element x in V. The set V together with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied.
A1. x+y=y+x for any x and y in V. A2. (x+y)+z=x+(y+z) for any x, y, z in V. A3. There exists an element 0 in V such that x+0=x for
each x ∈V. A4. For each x ∈V, there exists an element –x in V such
multiplication and addition operation ,is not a vector
space. Which of the eight axioms fail to hold?
The Vector Space C[a, b]
C[a,b] denote the set of all real-valued functions that are defined and continuous on the closed interval [a,b].
Chapter 3
Vector Spaces
The operations of addition and scalar multiplication are used in many contexts in mathematics. Regardless of the context, however, these operations usually obey the same set of algebra rules. Thus a general theory of mathematical systems involving addition and scalar multiplication will have application to many areas in mathematics.
(x1, x2 ) (x1,x2 )
(x1, x2 ) ( y1, y2 ) (x1 y1, 0)
We use the symbol to denote the addition operation
for this system avoid confusion with the usual addition x+y of row vectors. Show that S, with the ordinary scalar
Multiplication defined in the usual way.
Example Let S be the set of all ordered pairs of real
numbers. Define scalar multiplication and addition on S by
The closure properties of the two operations:
C1. If x ∈V and α is a scalar, then αx ∈V . C2. If x,y ∈V , then x+y ∈V .
Example Let W={(a,1) a real} with addition and scalar