线性代数第一章初等变化和秩英文版

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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 =。

线性代数课件第一章

0 0 0 0 0 0 ≠ ( 0 0 0 0) . 0 0 0
1 0 (5)单位矩阵单位矩阵 0 1 E = En = L L O 0 0
称为单位矩阵（单位阵）称为单位矩阵（或单位阵）. 单位矩阵
L 0 O L 0 L L L 1
a11 a 21 A= L a m1
简记为
a12 a 22 L am1
L a1 n L a2n L L L a mn
矩阵A的 (m, n)元
A = Am×n = (aij )m×n = (aij ).
这m × n个数称为 A的元素 ,简称为元 . 简称为元
a11 x1 + a12 x2 + L + a1n xn = b1 a x + a x +L + a x = b 21 1 22 2 2n n 2 LLLLLLLLLLLL am1 x1 + am 2 x2 + L + amn xn = bm
, , 系数 aij ( i =1,2,L m, j =1,2,L n) , 常数项 bi (i = 1,2,L,n)
全为1 全为
(6)方阵方阵主对角线
a11 a12 a21 a22 A= L L 副对角线 an1 an1
简记为
L a1n L a2 n L L L ann
n× n
矩 A 阵的
( n, n) 元
A = An× n = ( aij )
.
矩阵的转置
a11 a 21 A= L a m1
定义3 如果矩阵A经过有限次的初等变换变成B 定义3 如果矩阵A经过有限次的初等变换变成B，就称矩阵A与矩阵B等价，就称矩阵A与矩阵B等价，记作 A ~ B ．矩阵之间的等价具有自反性、对称性和传递性．矩阵之间的等价具有自反性、对称性和传递性．例如用矩阵的初等行变换解线性方程组

线性代数及其应用术语要点中英对照

1 / 12
[P50]
free variable 至少有一个自由变量注：结合简化阶梯形采用反证法轻松搞定！ Additionally, 此外：if r = #{pivot positions}, p = #{free variables}, n = #{variables} then r+p = n, #{} - number of {ζ} （ζ 的个数）注：看简化阶梯形 6. 非齐次线性方程组解的结构定理（ Structure of Solution Set of Nonhomogeneous System）关键词：nonhomogeneous system 非齐次线性方程组[P50]; Let v0 be a solution of a nonhomogeneous system Ax = b. Let H be the set of general solutions of the corresponding homogeneous system Ax = 0. Suppose the solution set of Ax = b is S Then S = H + v0 如果 v0 是非齐次线性方程组 Ax = b 的一个解，H 是对应齐次线性方程组 Ax = 0 的通解。（Ax = 0 也称为 Ax = b 的导出组）则 Ax = b 的通解是 S = H + v0
关键词： m 维空间 [P28]; subset of spanned (or generated) by v1, v2,…vp 由 v1, v2,…vp 张成(或生成的)的的子空间[P35]; 中的每一个向量 b，线性方程组 Ax = 1) For each b in , the system Ax = b has a solution.对于 b 都有一个解 2) Each b in is a linear combination of the columns of A. 中的每一个向量 b 都是矩阵 A 的列向量的线性组合 3) The columns of A span 矩阵 A 的列向量生成 4) The matrix A has a pivot position in every row. 矩阵 A 每一行都有一个主元位置注：1）- 3）根据定义显然成立；4）可用定理 2 采用反证法 10. 补充齐次方程组基础解系定理（ Additional Theorem of basic solutions of a homogenous linear system）关键词：basic solutions (基础解系) [讲义 P17 定理 5.3]

线性代数英文讲义

Chapter 4 Linear Transformations In this chapter, we introduce the general concept of linear transformation from a vector space into a vector space. But, we mainly focus on linear transformations from n R to m R.§1 Definition and ExamplesNew words and phrasesMapping 映射Linear transformation 线性变换Linear operator 线性算子Dilation 扩张Contraction 收缩Projection 投影Reflection 反射Counterclockwise direction 反时针方向Clockwise direction 顺时针方向Image 像Kernel 核1.1 Definition★Definition A mapping(映射) L: V W is a rule that produces a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set.★Definition A mapping L from a vector space V into a vector space W is said to be a linear transformation（线性变换）if(1) 11221122(v v )(v )(v )L L L αααα+=+for all 12v ,v V ∈ and for all scalars 1α and 2α. (1) is equivalent to(2) 1212(v v )(v )(v )L L L +=+ for any 12v ,v V ∈ and(3) (v)(v)L L αα= for any v V ∈ and scalar α.Notation: A mapping L from a vector space V into a vector space W is denotedL: V →WWhen W and V are the same vector space, we will refer to a linear transformation L: V →V as a linear operator on V . Thus a linear operator is a linear transformation that maps a vector space V into itself.1.2 Linear Operators on 2R1. Dilations(扩张) and Contractions Let L be the operator defined byL(x)=k xthen this is a linear operator. If k is a positive scalar, then the linear operator can be thought of as a stretching or shrinking by a factor of k.120(x)x 0x k L A x k ⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭2. Projection （投影）onto the coordinate axes.L(x)=11e x 1210(x)x 00x L A x ⎛⎫⎛⎫== ⎪⎪⎝⎭⎝⎭ L(x)=22e x 1200(x)x 01x L A x ⎛⎫⎛⎫== ⎪⎪⎝⎭⎝⎭3. Reflections (反射) about an axis Let L be the operator defined byL(x)=12(,)T x x -, then it is a linear operator. The operator L has theeffect of reflecting vectors about the x-axis. 1210(x)x 01x L A x ⎛⎫⎛⎫== ⎪ ⎪-⎝⎭⎝⎭Reflecting about the y-axis L(x)=12(,)T x x -, 1210(x)x 01x L A x -⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭4. RotationsL(x)=21(,)T x x -, L has the effect of rotating each vector by 90 degrees in the counterclockwise direction （逆时针方向）.1201(x)x 10x L A x -⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭1.3 Linear Transformations from n R to m RIf A is an mxn matrix, then we can define a linear transformation A L from n R to m R by()A L X AX =It is easy to verify that the mapping above is linear. In the next section, we will see that any linear transformation from n R to m R must be of this form.1.4 The Image and Kernel★Definition Let L: n R →m R is a linear transformation. The kernel （核）of L denoted ker(L), is defined by ker(L)={}v |(v)0W V L ∈=★Definition Let L: n R →m R is a linear transformation and let S be a subspace of V . The image （像）of S, denoted L(S), is defined by L(S)= {}w |w (v) for some v m n R L R ∈=∈The image of the entire vector space, L(V), is called the range （值域）of L.Theorem 4.1.1 If L: n R →m R is a linear transformation and S is a subspace of n R , then (i) ker(L) is a subspace of n R . (ii) L(S) is a subspace of m R .Assignment for section 1, chapter 4Hand in: 3, 4, 17, 20,Not required : 8, 10, 11, 15, 16, 19, 25§2 Matrix Representations of Linear TransformationsNew words and phrasesMatrix representation 矩阵表示 Formal multiplication 形式乘法 Similarity 相似性2.1 Matrix Representation of Linear TransformationsIn section 1 of this chapter, the examples of linear transformations can be represented by matrices. In general, a linear transformation can be represented by a matrix.If we use the basis E=[12u ,u ,,u n ] for U and the basisF=[12v ,v ,,v m ] for V , and L: U → V .If u is a vector in U, then1122u u u u n n x x x =+++ (in U) |→ 1122L(u)v v v m m y y y =+++ (in V)The linear transformation L is determined by the change of the coordinate vectors:12n x x x ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭→12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭Assume that1122(u )v v v j j j mj m L a a a =+++, j=1, 2, …, nFormally,12[(u ),(u ),,(u )]n L L L =12[v ,v ,,v ]m 111212122212n n m m mn a a a a a a a a a ⎛⎫ ⎪ ⎪⎪⎪⎝⎭Write 12[(u ),(u ),,(u )]n L L L as linear combinations of 12[v ,v ,,v ]m , then consequently, A is obtained.Then L(u) = 1122(u u u )n n L x x x +++ =12[v ,v ,,v ]m 12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭(formal multiplication)1122L(u )L(u )L(u )n n x x x =+++=nj=1(u )j j x L ∑1212[L(u ),L(u ),,L(u )]n n x xx ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭(formal multiplication)==11(v )n mj ij i j i x a =∑∑=i=11()v m nij j i j a x =∑∑=12[v ,v ,,v ]m 111212122212n n m m mn a a a a a a a a a ⎛⎫ ⎪ ⎪ ⎪⎪⎝⎭12n x x x ⎛⎫ ⎪ ⎪⎪ ⎪⎝⎭Hence12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭=111212122212n n m m mn a a a aa a a a a ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭12n x x x ⎛⎫ ⎪ ⎪⎪ ⎪⎝⎭Thus, y=A x is the coordinate vector of L (u) with respect toF =[12v ,v ,,v m ]. y=A x is called the matrix representation of thelinear transformation. A is called the matrix representing L relative to thebases E and F. A is determined by the following equations.12[(u ),(u ),,(u )]n L L L =12[v ,v ,,v ]m 111212122212n n m m mn a a a a a a a a a ⎛⎫ ⎪ ⎪⎪⎪⎝⎭We have established the following theorem. Theorem 4.2.2 If E=[12u ,u ,,u n ] is an ordered basis for U andF=[12v ,v ,,v m ] is an ordered basis for V , then corresponding to eachlinear transformation L:U →V there is an mxn matrix A such that [()][]F E L u A u = for each u in U.A is the matrix representing L relative to the ordered bases E and F. In fact, a [(u )]j j F L =.2.2 Matrix Representation of L: n R →m RIf U=n R , V=m R , then we have the following theorem.Theorem 4.2.1 If L is a linear transformation mapping n R into m R , there is an mxn matrix A such thatL (x)=A xfor each x n R ∈. In fact, the jth column vector of A is given by12((e ),(e ),,(e ))n A L L L =Proof If we choose standard basis 12[e ,e ,,e ]n for n R and thestandard basis 12[e ,e ,,e ]m for m R ,L(x)= 1122(e e e )n n L x x x +++=12(e ,e ,,e )m 12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭=12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭1122L(e )L(e )L(e )n n x x x =+++=nj=1(e )j j x L ∑1212(L(e ),L(e ),,L(e ))n n x x x ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭And let A=()ij a =()12a ,a ,,a n 12(L(e ),L(e ),,L(e ))n =If 1122x e e e n n x x x =+++, then L(x)=Ax.A is referred to as the standard matrix representation (标准矩阵表示) of L.( A representation with respect to the standard basis.)Example 1 (example 1 on page 186) Determine the standard matrix representation of L.Define the linear transformation L:3R 2R by1223(x)(,)T L x x x x =++ for each 123x (,,)T x x x = in 3R , find the linear standard representation of L.Solution: Find 123L(e ),L(e ),L(e ). Then 123110(L(e ),L(e ),L(e ))011A ⎛⎫== ⎪⎝⎭Example 2 rotation by an angle θLet L be the linear transformation operator on 2R that rotates each vector by an angle θ in the counterclockwise direction. We can see that1e is mapped to (cos ,sin )T θθ, and 2e is mapped to (sin ,cos )T θθ-.1(e )(cos ,sin )T L θθ=, 2(e )(sin ,cos )T L θθ=-The matrix A representing the transformation will be12cos sin (L(e ),L(e ))sin cos A θθθθ-⎛⎫==⎪⎝⎭To find the matrix representation A for a linear transformation L n R→ m R w.r.t. the bases E=[12u ,u ,,u n ] and F=[12b ,b ,,b m ], wemust represent each vector 1122(u )b b b j j j mj m L a a a =+++. The followingtheorem shows that determining this representation is equivalent to solving the linear system Bx=(u )j L , where (u )j L is regarded as a column vector in m R .Theorem 4.2.3 Let E =[12u ,u ,,u n ] and F =[12b ,b ,,b m ] beordered bases for n R and m R , respectively. If L : n R → m R is a linear transformation and A is the matrix representing L with respect to E and F , then112((u ),(u ),,(u ))n A B L L L -=where B =(12b ,b ,,b m ).Proof L(u) = 1122(u u u )n n L x x x +++ =12(b ,b ,,b )m 12m y y y ⎛⎫⎪ ⎪ ⎪ ⎪⎝⎭1122L(u )L(u )L(u )n n x x x =+++=nj=1(u )j j x L ∑1212(L(u ),L(u ),,L(u ))n n x xx ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭12(b ,b ,,b )m 12m y y y ⎛⎫ ⎪ ⎪ ⎪ ⎪⎝⎭1212(L(u ),L(u ),,L(u ))n n x x x ⎛⎫ ⎪ ⎪= ⎪ ⎪⎝⎭The matrix B is nonsingular since its column vectors form a basis form R . Hence, 112((u ),(u ),,(u ))n A B L L L -=12((u ),(u ),,(u ))n L L L is the matrix representing L relative to the bases [12u ,u ,,u n ] and [12e ,e ,,e m ]. B is the transition matrixcorresponding to the change of basis from [12b ,b ,,b m ] to[12e ,e ,,e m ].Corollary 4.2.4 If A is the matrix representing the linear transformation L:n R m R with respect to the bases12m [b ,b ,,b ]1B -12m [e ,e ,,e ]12n [u ,u ,,u ] A12((),(),,())n L u L u L uBE=[12u ,u ,,u n ] and F=[12b ,b ,,b m ]then the reduced row echelon form of1212(b ,b ,,b |(u ),L(u ),,L(u ))m n Lis (I |A )Proof 1212(b ,b ,,b |(u ),L(u ),,L(u ))m n L =(B|BA), which is rowequivalent to (I|A).Examples Finding the matrix representing L Example 3 on page 188Let L be a linear transformation mapping 3R into 2R defined by 11232L(x)b ()b x x x =++. Find the matrix A representing L with respect to the ordered bases 123[e ,e ,e ] and 12[b ,b ], where 11b 1⎛⎫= ⎪⎝⎭, 21b 1-⎛⎫= ⎪⎝⎭Solution:Method 1. Represent 123[e ,e ,e ] in terms of 12[b ,b ] Method 2. 112123(b ,b )((e ),(e ),(e ))A L L L -=1111111/21/2111100111111/21/2111011A ------⎛⎫⎛⎫⎛⎫⎛⎫⎛⎫=== ⎪ ⎪ ⎪⎪ ⎪-⎝⎭⎝⎭⎝⎭⎝⎭⎝⎭Method 3 Applying row operations. 12123(b ,b |(e ),(e ),(e ))L L L Example 4 on page 188Let L be a linear transformation mapping 2R into itself defined by 1212L(b b )()b 2b αβαββ+=++, where 12[b ,b ] is the ordered basisdefined in example 3. Find the matrix A representing L with respect to12[b ,b ].Solution: Use three methods as in example 3. Example 6 on page 190Determine the matrix representation of L with respect to the given bases. Let L: 2R 3R be the linear transformation defined by 21212L(x)(,,)T x x x x x =+-Find the matrix representation of L with respect to the ordered bases12[u ,u ] and 123[b ,b ,b ], where12u (1,2),u (3,1)T T ==123b (1,0,0),b (1,1,0),b (1,1,1)T T T ===Assignment for section 2, chapter 4Hand in: 2, 6, 8, 16, 20 Not required: 9—15, 17, 19§3 SimilarityLet L be a linear operator on V , E=[12v ,v ,,v n ] be an orderedbasis for V , A is the matrix representing L with respect to the basis E. 1122u v v v n n x x x =+++, 1122L(u)v v v n n y y y =+++1211221212(v )v v v [v ,v ,,v ][v ,v ,,v ]a j j j j j nj n n n j nj a a L a a a a ⎛⎫ ⎪ ⎪=+++== ⎪ ⎪ ⎪⎝⎭y=Ax F=[12w ,w ,,w n ]1122u w w w n n c c c =+++, 1122L(u)w w w n n d d d =+++1211221212(w )w w w [w ,w ,,w ][w ,w ,,w ]b j jj j j nj n n n j nj b b L b b b b ⎛⎫ ⎪ ⎪=+++== ⎪ ⎪ ⎪⎝⎭d=BcLet the transition matrix corresponding the change of basis from F=[12w ,w ,,w n ] to [12v ,v ,,v n ]Then x=Sc, y=Sd, 11y x S BS --= 1y x SBS -= or 1A SBS -=Hence, we have established the following theorem.Theorem 4.3.1 Let E=[12v ,v ,,v n ] and F=[12w ,w ,,w n ] betwo ordered bases for a vector space V , and let L be a linear operator onn R . Let S be the transition matrix representing the change from F to E. IfA is the matrix representing L with respect to E, andB is the matrix representing L with respect to F, then 1B S AS -=.★Definition Let A and B be nxn matrices. B is said to be similar to A if there is a nonsingular matrix S such that 1B S AS -=.Example 2 (on page 204)Example Let L be the linear operator on 3R defined by L(x)=Ax, whereV VV VBasis E=12n [v ,v ,,v ] Su →L(u)u →L(u)Ax=yBc=dS -1Coordinate vector of L(u): dBasis FCoordinate vector of L(u) :y Basis ECoordinate vector of u: xCoordinate vector of u: c Basis E=12n [w ,w ,,w ] x=Sc y=Sd220112112⎛⎫ ⎪ ⎪ ⎪⎝⎭. Thus the matrix A represents L with respect to the standard basis for 3R . Find the matrix representing L with respect to the basis [123y ,y ,y ], where 1y (1,1,0)T =-, 2y (2,1,1)T =-, 3y (1,1,1)T =. SolutionD=000010004⎛⎫⎪⎪ ⎪⎝⎭is the matrix representing L w.r.t the basis [123y ,y ,y ],. Or, we can find D using 1D Y AY -= 11x ()x=()x n n n A YDY YD Y --=Using this example to show that it is desirable to find as simple as a representation as possible for a linear operator. In particular, if the operator can be represented by a diagonal matrix, this is usually preferred representation. It makes the computation of Dx and x n D easier.Assignment for section 3, chapter 4Hand in: 2, 3, 4, 8, 10, 15 Not required 5, 6。

线代名词中英文对照

《线性代数》英文专业词汇序号英文中文1LinearAlgebra线性代数2determinant行列式3row行4column列5element元素6diagonal对角线7principaldiagona主对角线8auxiliarydiagonal次对角线9transposeddeterminant转置行列式10triangulardeterminants三角行列式11thenumberofinversions逆序数12evenpermutation奇排列13oddpermutation偶排列14parity奇偶性15interchange互换16absolutevalue绝对值17identity恒等式18n-orderdeterminantsn阶行列式19evaluationofdeterminant行列式的求值20Laplace’sexpansiontheorem拉普拉斯展开定理21cofactor余子式22Algebracofactor代数余子式23theVandermondedeterminant范德蒙行列式24bordereddeterminant加边行列式25reductionoftheorderofadeterminant降阶法26methodofRecursionrelation递推法27induction归纳法28Cramer′s rule克莱姆法则29matrix矩阵30rectangular矩形的31thezeromatrix零矩阵32theidentitymatrix单位矩阵33symmetric对称的序号英文中文34skew-symmetric反对称的35commutativelaw交换律36squareMatrix方阵37amatrixoforder m×n矩阵m×n38thedeterminantofmatrixA方阵A的行列式39operationsonMatrices矩阵的运算40atransposedmatrix转置矩阵41aninversematrix逆矩阵42anconjugatematrix共轭矩阵43andiagonalmatrix对角矩阵44anadjointmatrix伴随矩阵45singularmatrix奇异矩阵46nonsingularmatrix非奇异矩阵47elementarytransformations初等变换48vectors向量49components分量50linearlycombination线性组合51spaceofarithmeticalvectors向量空间52subspace子空间53dimension维54basis基55canonicalbasis规范基56coordinates坐标57decomposition分解58transformationmatrix过渡矩阵59linearlyindependent线性无关60linearlydependent线性相关61theminorofthe k thorderk阶子式62rankofaMatrix矩阵的秩63rowvectors行向量64columnvectors列向量65themaximallinearlyindependentsubsystem最大线性无关组66Euclideanspace欧几里德空间67Unitaryspace酉空间序号英文中文68systemsoflinearequations线性方程组69eliminationmethod消元法70homogenous齐次的71nonhomogenous非齐次的72equivalent等价的73component-wise分式74necessaryandsufficientcondition充要条件75incompatiable无解的76uniquesolution唯一解77thematrixofthecoefficients系数矩阵78augmentedmatrix增广矩阵79generalsolution通解80particularsolution特解81trivialsolution零解82nontrivialsolution非零解83thefundamentalsystemofsolutions基础解系84eigenvalue特征值85eigenvector特征向量86characteristicpolynomial特征多项式87characteristicequation特征方程88scalarproduct内积89normedvector单位向量90orthogonal正交的91orthogonalization正交化92theGram-Schmidtprocess正交化过程93reducingamatrixtothediagonalform对角化矩阵94orthonormalbasis标准正交基95orthogonaltransformation正交变换96lineartransformation线性变换97quadraticforms二次型98canonicalform标准型99thecanonicalformofaquadraticform二次型的标准型100themethodofseparatingperfectsquares配完全平方法101thesecond-ordercurve二次曲线102coordinatetransformation坐标变换。

(完整版)线性代数笔记

等行变换，则得到的是。
对于第二类的可先转化为第一类的，即由
两边转置得
按上例的方法求出进而求出 X
二.初等变换的性质
定理 2.5.1 设线性方程组的增广矩阵经有限次的初等行变换化为，则以与
为增广矩阵的方程组同解。定理 2.5.2 任何矩阵都可以经有限次初等行变换化成行最简形式，经有限次初等变换（包括行及列）化成等价标准形。且其标准形由原矩阵惟一确定，而与所做的初等变换无
3、矩阵的乘法设 A=（aij）m×n，B=（bjk）n×l，则 A*B=C=（cik）m×l 其中 C=Σaijbjk（j=1，n）注意；两个矩阵相乘必须第一个矩阵的列数等于第二个矩阵的行数；矩阵乘法不满足交换律，即 AB 不一定等于 BA；矩阵乘法有零因子，即 A≠0（零矩阵），B≠0（零矩阵），但有可能 A*B=0（零矩阵）矩阵的乘法适合以下法则：（1）结合律：（AB）C=A（BC）（2）分配律（A+B）C=AC+BC
hing at a time and All things in their being are good for somethin
此处 0 表示与 A 同型的零矩阵，即 A=（aij）m×n ，0=0m×n （4）矩阵 A=（aij）m×n，规定-A=（-aij）m×n，（称之为 A 的负矩阵），则有 A+（-A）=（A）+A=0
如果 n 个未知数，n 个方程的线性方程组的系数行列式 D≠0,则方程组
定理 1.4.3 如果 n 个未知数 n 个方程的齐次方程组的系数行列式 D≠0，则该方程组只有零解，没有非零解。推论如果齐次方程组有非零解，则必有系数行列式 D=0。
第二章矩阵
一、矩阵的运算

LINEAR ALGEBRA 线性代数课文翻译

4 LINEAR ALGEBRA 线性代数“Linear algebra” is the study of linear sets of equations and their transformation properties. 线性代数是研究方程的线性几何以及他们的变换性质的Linear algebra allows the analysis of rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering.线性代数也研究空间旋转的分析，最小二乘拟合，耦合微分方程的解，确立通过三个已知点的一个圆以及在数学、物理和机械工程上的其他问题The matrix and determinant are extremely useful tools of linear algebra.矩阵和行列式是线性代数极为有用的工具One central problem of linear algebra is the solution of the matrix equation Ax = b for x. 线性代数的一个中心问题是矩阵方程Ax=b关于x的解While this can, in theory, be solved using a matrix inverse x = A−1b，other techniques such as Gaussian elimination are numerically more robust.理论上，他们可以用矩阵的逆x=A-1b求解，其他的做法例如高斯消去法在数值上更鲁棒。

线性代数_常用单词词组_中英对照

IndexEconomics 经济学435,439 Eigencourse 457,458Eigenvalue 特征值283,287,374,499 Eigenvalue changes 特征值变换439Eigenvalues of 的特征值284,294,300 Eigenvalues of 的特征值297Eigenvalues of 的特征值362Eigenvector basis 基底的特征向量399Eigenvectors 特征向量283,287,374Eigshow 290,368Elimination 消元法45-66,83,86,135 Ellipse 椭圆290,346,366,382 Energy 能量343,409Engineering 工程409,419Error 误差211,218,219,225,481,483 Error equation 误差方程477Euler angles 欧拉角474Euler’s formula 欧拉公式311,426,430,497Even 偶数113,246,258,452 Exponential 指数的314,319,327FFactorization 因式分解95,110,235,348,370,374Householder reflections 镜像变换237,469,472 Hyperplane 超平面30,42IIll-conditioned matrix 病态矩阵371,473,474 Imaginary 虚数289Independent 独立的26,27,134,168,200,300 Initial value 初值313Inner product 积11,56,108,448,502,506 Input and output basis 基底输入输出399Integral 积分24,385,386Interior point method 点法445Intersection of spaces 交空间129,183Inverse matrix 逆矩阵24,81,270Inverse of的逆82Invertible 可逆的86,173,200,248 Iteration 迭代481,482,484,489,492 JJacobi 雅可比481,483,485,489 Jordan form 约当型356,357,358,361,482 JPEG 364,373KKalman filter 卡尔曼滤波器93,214Kernel 核377,380Kirchhoff’s Laws 基尔霍夫定律143,189,420,424-427 Krylov 克雷洛夫491,492Lnorm 和数225,480Lagrange multiplier 拉格朗日乘子445Lanczos method 兰索斯方法490,492LAPACK 线性代数软件包98,237,486Leapfrog method 跳步法317,329Least squares 最小平方218,219,236,405,408,453 Left nullspace左零空间184,186,192,425Left-inverse 左逆的81,86,154,405Length 长度12,232,447,448,501 Line 线34,40,221,474Line of springs 线弹簧411Linear bination 线性组合1,3Linear equation 线性方程23Linear programming 线性规划440Linear transformation 线性变换44,375-398Linearity 线性关系44,245,246Linearly independent 线性独立26,134,168,169,200 LINPACK 线性系统软件包465Loop 环路307,425,426Lower triangular 下三角9598,100,474Lucas numbers 卢卡斯数306MMaple 38,100Mathematica 38,100MATLAB 17,37,237,243,290,337,513 Matrix(see full page 570) 矩阵22,384,387Matrix exponential 矩阵指数314,319,327Matrix multiplication 矩阵乘法58,59,67,389Matrix notation 矩阵记号37Matrix space 矩阵空间121,122,175,181,311Matrix 矩阵-1,2,-1 matrix -1，2，-1矩阵106,167,261,265,349,374,410,48Adjacency, 邻接矩阵74,80,311,369All-ones 全1矩阵251,262,307,348 Augmented 增广矩阵60,84,155Band 带状矩阵99,468,469Block 分块矩阵70,94,115,266,348Circulant 循环矩阵507,515. .- -可修编.。

山东大学《线性代数》课件01-5矩阵的初等变换与矩阵的秩

2
3
1 3 0 6
0 0
8 2
2 12 1 4
1 4 1 3 1 4
2 12 0 6 4 4
8
2
0 9 6 6
1 4 4 4 0 0
r( A) 2
1 2 3 4 1 2 3 4
2.B
1 13
0 1 2
1 1 0
2 05
0 0 0
2 7 0
2 10 3
2 192
1
0 0 0
2 1 7 0
3 1 10 3
4 1 192
1
0
0 0
2 1 0 0
3 1 3 3
4
1
95
1 2 3 4
0 00
1 0 0
1 3 0
1
45
r(B) 4
1 A 4
2 t
2 3
3 12
t为何值时， r( A) 3？
3
1
1
9
1 A 0
2 t 8
a1n
ai1
ka j1
ai2 kaj2
ain
kajn
B
a j1
a j2
a jn
am1
am2
amn
由此可以推出:
r( A) r(B) r( A) r(B) r( A) r(B)
例:求矩阵的秩:
2 3 1.A 2 12 1 3
1 3
A 2 12
r1r3
1 2 2 3
1
2
2 3
B 4 3 3 12 0 11 11 0
3 1 1 9 0 7 7 0
1 0
2 1
2 1

线性代数英文课件：ch1_1 Definition

Linear Algebra
Math. Dept., Wuhan University of Technology
Textbook：工程数学-线性代数,第五版，同济大学数学系编，高等教育出版社，2011 References:
➢线性代数（第7版），S.J.Leon，机械工业出版社，2007 （Linear Algebra With Applications） ➢经济数学：线性代数，吴传生等编，高等教育出
Math. Dept., Wuhan University of Technology
Linear Algebra History
➢Leibniz introduced the definition of determinant in 17th century。 (日本数学家关孝和Seki Kowa将其概念称为“行列式”)
Example 1.
2 -1
Evaluate
.
34
23
Example 2. Evaluate
.
-1 4
Example 3.
23
Evaluate
.
15
Math. Dept., Wuhan University of Technology
Sec.1 Determinants of Order 2 and 3
24 8 4 16 4
2 10 Example 5. Evaluate D3 1 1 4
3 2 5
Math. Dept., Wuhan University of Technology
3.Permutations &Number of Inversions
? How to generalize the definitions of 2×2 and 3×3 determinants to n×n determinants? Analyzing formula (1),we can get : (1) It’s the algebraic sum of six (which is exactly the

线性代数-矩阵的初等变换

求解未知量
根据行最简形式的矩阵，直接求解出未知量的值。
案例分析：具体求解过程展示
案例一
01
简单线性方程组求解过程展示，包括构造增广矩阵、进行初等
变换和求解未知量等步骤。
案例二
02
复杂线性方程组求解过程展示，涉及更多未知量和更复杂的增
广矩阵，展示如何利用初等变换求解该类问题。
案例三
03
含参数线性方程组求解过程展示，通过引入参数，展示如何对
含参数的线性方程组进行求解和分析。
04 初等变换在矩阵秩计算中应用
矩阵秩定义及性质
矩阵秩定义：矩阵A中不等于0的子式的最大阶数称为
矩阵A的秩，记作r(A)。
矩阵秩的性质
矩阵的秩是非负的，且等于其行秩或列秩。
若矩阵A可逆，则r(A)=n，其中n为A的阶数。
若矩阵A为0矩阵，则 r(A)=0。
初等变换与矩阵的等价关系
通过初等变换，我们可以得到与原矩阵等价的矩阵。这种等价关系在线性代数中具有重要意义，它揭示了矩阵之间的一种本质联系。
初等变换在求解线性方程组中的应用
通过对方程组的增广矩阵进行初等变换，我们可以将方程组化为简化阶梯形式，从而方便地求出方程组的解。
对未来研究方向和趋势展望
深入研究初等变换的性质和应用
条件
01
非零行的首非零元为1；
02
首非零元所在列的其他元素全为零。
03
性质
最简形矩阵是唯一的；
对于任意行阶梯形矩阵，总可
04
05
以通过初等行变换化为最简形
矩阵。
06
行阶梯形与最简形矩阵，二者都可以通过初等行变换得到。
区别
行阶梯形矩阵只要求非零行的首非零元所在列的上三角元素全为零，而最简形矩阵还要求非零行的首非零元为1，且所在列的其他元素全为零。因此，最简形矩阵比行阶梯形矩阵具有更简洁的形式。

线代英文词汇

线代英文词汇adjontof matrix A A 的伴随矩阵augmented matrix A 的增广矩阵basic solution set 基础解系Cauchy-Schwarz inequality柯西-许瓦兹不等式characteristic equation 特征方程characteristic polynomial 特征多项式coffcient matrix 系数矩阵cofactor 代数余子式cofactor expansion 代数余子式展开column vector 列向量commuting matrices 交换矩阵consistent linear syste相容线性方程组Cramer’s rule 克莱姆法则Cross- product term 交叉项Determinant 行列式Diagonal entries 对角元素Diagonal matrix 对角矩阵Dimension of a vector space V 向量空间V的维数eigenspace 特征空间eigenvalue 特征值eigenvector 特征向量eigenvector basis 特征向量的基elementary matrix 初等矩阵elementary row operations行初等变换full rank 满秩fundermental set of solution基础解系grneral solution 通解Gram-Schmidt process施密特正交化过程homogeneous linear equations齐次线性方程组identity matrix 单位矩阵inconsistent linear system不相容线性方程组indefinit quatratic form 不定二次型inner product 内积inverse of matrix A 逆矩阵linear combination 线性组合linearly dependent 线性相关linearly independent 线性无关linear transformation 线性变换lower triangular matrix 下三角形矩阵main diagonal of matrix 矩阵的主对角matrix 矩阵negative definite quaratic form负定二次型negative semidefinite quadratic form半负定二次型nonhomogeneous equations非齐次线性方程组nonsigular matrix 非奇异矩阵normalizing vector V 规范化向量orthogonal basis 正交基orthogonal decomposition 正交分解orthogonally diagonalizable matrix矩阵的正交对角化orthogonal matrix 正交矩阵orthogonal set 正交向量组orthonormal basis 规范正交基orthonomal set 规范正交向量组partitioned matrix 分块矩阵positive definite matrix 正定矩阵positive definite quatratic form正定二次型quatratic form 二次型rank of matrix A 矩阵A的秩reduced echelon matrix 最简梯形阵row vector 行向量set spanned by { } 由向量{ }所生成similar matrices 相似矩阵similarity transformation 相似变换singular matrix 奇异矩阵solution set 解集合standard basis 标准基standard matrix 标准矩阵I submatrix 子矩阵subspace 子空间symmetric matrix 对称矩阵trace of matrix A 矩阵A 的迹transpose of A 矩阵A的转秩unit vector 单位向量upper triangular matrix 上三角形矩阵vandermonde matrix 范得蒙矩阵vector space 向量空间zero subspace 零子空间zero vector 零空间。

(完整word版)线性代数课程专业词汇表

线性代数课程专业词汇表英文单词或词组中文翻译书中出现页码Linear equation 线性方程 1Linear system(s) 线性方程组 1Consistent 有解 2Inconsistent 无解 2Solution set of linear system 线性方程组的解集合 2Equivalent systems 等价的线性方程组 3Row operations 行变换 5Strict triangular form 严格三角形式 5Back substitution 回代法 6Equivalent systems 等价的线性方程组 6Coefficient matrix 系数矩阵 7Coefficient matrix 系数矩阵 7Augmented matrix 增广矩阵 8Pivot 主元 8Free variables 自由未知量 14Lead variables 前变量 14Gaussian elimination 高斯消元法 15Overdetermined linear system 方程个数超过未知数个数的方程组 15 Row echelon form 行阶梯形 15Underdetermined linear system 方程个数低于未知数个数的方程组 17 Gauss-Jordan reduction 高斯-若当归纳法 18Reduced row echelon form 减少的行阶梯形 18Homogeneous linear system 齐次线性方程组 22Homogeneous system 齐次线性方程组 22nontrivial solution 非零解 22Trivial solution 平凡解，全零解 22Matrix algebra 矩阵代数 30Scalars 常数 30Column vector(s) 列向量 31Euclidean n-space 欧几里得空间 31Row vector(s) 行向量 31Vector(s）向量 31Addition of matrices 矩阵加法 32Addition of matrices 矩阵加法 32Equality of matrices 矩阵相等 32Scalar multiplication for matrices 矩阵的数乘 32Scalar multiplication of matrices 矩阵的数乘 32Zero matrix 零矩阵 33Scalar product 内积 34Linear combination 线性组合 36Consistency Theorem 解的存在性定理 37Multiplication of matrices 矩阵乘法 38Identity matrix 单位矩阵 47Inverse matrix 逆矩阵 48Invertible matrix 可逆矩阵 48Nonsingular matrix 非奇异矩阵 48Singular matrix 奇异矩阵 49Transpose of a matrix 矩阵的转置 49Transpose of matrix 矩阵的转置 49Symmetric matrix 对称矩阵 51Symmetric matrix 对称矩阵 51Adjacency matrix 邻接矩阵 52Graph(s) 图 52Angle between vectors 向量的夹角 56Markov chain(s） Markov链 57Elementary matrix 初等矩阵 62Row equivalent 行等价 64Row equivalent matrices 行等价矩阵 64Diagonal matrix 对角矩阵 67Lower triangular 下三角 67Triangular factorization 三角分解 67Triangular matrix 三角形矩阵 67Triangular matrix 三角形矩阵 67Upper triangular 上三角 67Upper triangular matrix 上三角矩阵 67LU factorization LU分解 68Matrix factorizations 矩阵分解 68Partitioned matrices 分块矩阵 72Vandermonde matrix 范德蒙矩阵 72Block multiplication 分块乘法 74Inner product 内积 78Determinant(s) 行列式 90Cofactor 代数余子式 93Minor 余子式 93Cofactor expansion 代数余子式展开 94Determinant of matrix 矩阵的行列式 95Skew symmetric 反对称 105Adjoint of a matrix 伴随矩阵 106Cramer’s rule 克莱姆法则 107Cryptography 密码学 108Addition of vectors 向量的加法 119Closure properties 封闭性 119Vector space 向量空间 119Zero vector 零向量 119C[a,b] 区间[a,b]上的连续函数 120Isomorphism between vector spaces 向量空间的同构 123Subspace(s) 子空间 123Zero subsapce 零空间 125Nullspace 零化空间 127Nullspace of matrix 矩阵的零化空间 127Span 张成 128Spanning set 生成集 129Linearly dependent 线性相关 136Linearly independent 线性无关 136Basis 基 145Dimension 维数 147Finite dimensional 有限维 147Infinite dimensional 无限维 147Standard basis 标准基 150change of basis 基的变换 151Coordinate vector 坐标向量 152Transition matrix 过渡矩阵 155Coordinates 坐标 157Column space 列空间 162Column space of matrix 矩阵的列空间 162Rank of a matrix 矩阵的秩 162Rank of matrix 矩阵的秩 162Row space 行空间 162Row space of matrix 矩阵的行空间 162Nullity 零化度 164Rank-Nullity Theorem Rank-Nullity定理 164Left inverse 左可逆 170Right inverse 右逆 170Full rank 满秩 171Linear transformation(s) 线性变换 175Linear operator 线性算子 176Image 象 181Kernel 核 181Contraction 收缩 192Dilation 扩张 192Similarity 相似性 199Similar matrices 相似矩阵 202Trace 迹 206Angle between vectors 向量的夹角 211Euclidean length 欧几里得长度 211Distance in 2-space 2维空间的距离 212Cauchy-Schwarz inequality 柯西-施瓦兹不等式 213 Orthogonality 正交性 213Scalar projection 数量投影 214equation of plane 平面方程 215Nonmal vector 正规向量 215Angle between vectors 向量的夹角 216Pythagorean Law Pythagorean 定理 216Correlations 相关 219Correlation matrix 相关矩阵 221Covariance 协方差 222Covariance matrix 协方差矩阵 222Factor analysis 因子分析 222Fundamental subspaces 基本子空间 227Range of a matrix 矩阵的值域 227Direct sum 直和 229Least squares problem(s) 最小二乘法问题 234Projection onto column space 列空间上的投射 236Normal equations 正规方程 237Inner product space 内积空间 245Length in inner product spaces 内积空间中的长度 246 Orthogonal set(s) 正交集合 255Orthonormal set(s) 标准正交集 255orthonormal basis 标准正交基 256Orthonormal basis 标准正交基 256Orthogonal matrices 正交矩阵 258Orthogonal matrix 正交矩阵 258Approximation of functions 函数的逼近 264Fourier coefficients 傅里叶系数 266Fourier matrix 傅里叶矩阵 269Gram-Schmidt process Gram-Schmidt过程 274Dimension Theorem 维数定理 283Orthogonal polynomials 正交多项式 283Hermite polynomials Hermite 多项式 287Jacobi polynomials Jacobi多项式 287Lagrange’s interpolating formula Lagrange 插值公式 288 Gaussian quadrature 高斯求积 289Characteristic value(s) 特征值 301Characteristic vector 特征向量 301Eigenvalue 特征值 301Eigenvector 特征向量 301Characteristic equation 特征方程 302Characteristic polynomial 特征多项式 302Eigenspace 特征空间 302Nilpotent 幂零的 311Companion matrix 友矩阵 313Linear differential equations 线性微分方程 313Initial value problems 初值问题 314Diagonalizable matrix 可对角化的矩阵 326Distance in n-space n维空间的距离 332Complex matrix 复矩阵 346Hermite matrix Hermite 矩阵 346Unitary matrix 酉矩阵 347Unitary matrix 酉矩阵 347Normal matrices 正规矩阵 351Singular values 奇异值 356Conic sections 二次曲线部分 371Quadratic equation in n variables n个变量的二次方程 376 Quadratic form in n variables n个变量的二次型 376 Definite quadratic form 定二次型 378Indefinite quadratic form 不定二次型 378Negative definite matrix 负定矩阵 378Negative definite quadratic form 负定二次型 378 Negative semidefinite matrix 半负定矩阵 378Negative semidefinite quadratic form 半负定二次型 378 Positive definite matrix 正定矩阵 378Positive definite quadratic form 正定二次型 378Positive semidefinite matrix 半正定矩阵 378Positive semidefinite quadratic form 半正定二次型 378 Local maximum 极大值 382Local minimum 极小值 382Positive definite matrix 正定矩阵 384Leading principal submatrix 顺序主子矩阵 385 Nonnegative matrix 非负矩阵 392Nonnegative vector 非负向量 392Positive matrix 正矩阵 392Positive matrix 正矩阵 392Reducible matrix 可约矩阵 394Frobenius theorem Frobenius 定理 395Absolute error 绝对误差 411Relative error 相对误差 411Back substitution 回代法 419QR factorization QR分解 448。

线性代数

2
3
4
LP Decoding
LP Decoding for Non-Uniform Sources
LP Decoding for the Polya Contagion Channel
Exploiting Source Redundancy at the Decoder
Non-uniformity at the source can be exploited at the decoder. Assuming a systematic (n, k ) code C , it is possible to linearize the MAP decoding metric so as to exploit non-uniformity in an LP decoder: c ˆ = argmax P (c )P (y |c )
10
−1
10 PCE 10
−2
−3
10
−4
Standard LP Decoder Systematic LP Decoder Non−Systematic LP Decoder 10
−5
0.02
0.04
0.06
0.08
ρ
0.1
0.12
0.14
0.16
Figure: Source p1 = 0.9. Top two curves: regular systematic (200, 100) LDPC code. Bottom curve: regular (300, 100) LDPC code with the ﬁrst 100 (systematic) bits punctured.
A. Cohen, F. Alajaji, N. Kashyap, G. Takahara

线性代数术语中英文对照_第一章

1linear equation [P2] 线性方程coefficient [P2] 系数constant term [讲义‐P1] 常数（项）systems of linear equations (linear system) [P3] 线性方程组solution [P3] 解solution set [P3] 解集tuple [P3] 数组equivalent [P3] 等价equivalent system [P3] 等价系统，等价方程组 consistent [P4] 相容inconsistent [P4] 不相容consistent linear system 相容（有解）的线性方程组inconsistent linear system 不相容（无解）的线性方程组coefficient matrix [P5] 系数矩阵augmented matrix [P5] 增广矩阵elementary operations [P7] 初等变换elementary row operation [P7] 初等行变换row equivalent [P7] 行等价（矩阵） equivalent matrix [讲义‐P3] 等价矩阵Leading entry [P14] 先导元素entry of matrix 矩阵的元素echelon form (row echelon form) [P14] 阶梯形reduced echelon form (reduced row echelon form) [P14] 简化阶梯形Gaussian elimination [P14]脚注高斯消元法the elimination of variables 消元法row reduced [P15] 行化简pivot position [P16] 主元位置pivot column [P16] 主元列pivot [P17] 主元forward phase [P20] 自上（向下）阶段 backward phase [P20] 自下（向上）阶段basic variable [P20] 基本变元free variable [P20] 自由变元general solution [P21] 一般解，通解back‐substitution [P22] 回代法，侧转代入vector [P28] 向量，矢量scalar [P28] 数量，纯量，无向量 parallelogram rule [P30] 平行四边形法则linear combination [P32] 线性组合generated (spanned) [P35] 生成(张成)product [P45] 乘积homogeneous systems [P50] 齐次系统，齐次线性方程组 trivial solution [P50] 平凡解，零解nontrivial solution [P51] 非平凡解，非零解 parametric vector form [P52] 参数向量形式 nonhomogeneous systems [P50] 非齐次线性方程组linear independent [p65] 线性相关linearly independent [P65] 线性无关linear transformation [P73] 线性变换domain [P73] 值域shear transformation [P76] 剪切变换contraction [P77] 压缩变换dilation [P77] 拉伸变换。

高等数学【线性代数】英文版课件1

1 2 3 4
dy 2 dx + y = x d2 y = −k2 y dx2 2y 5 d3 y + d 2 + cos x = dx3 dx dy sin dx + tan−1 y = 1
0
Ordinary Differential Equations Lecture Notes
Deﬁnition (1.2.3) The order of the highest derivative occurring in a differential equation is called the order of the differential equation. In Example 1.2.2
Ordinary Differential Equations Lecture Notes
School of Physical and Mathematical Sciences Nanyang Technological University
August 2010
Ordinary Differential Equations Lecture Notes
Ordinary Differential Equations Lecture Notes
1.2. Basic Ideas and Terminology
Begin with a very general deﬁnition of a differential equation. Deﬁnition (1.2.1) A differential equation is an equation involving one or more derivatives of an unknown function. Examples (1.2.2) The following are all differential equations.

线性代数第一章初等变化和秩英文版

线性代数 英文讲义

线性代数课件 第一章

线性代数及其应用术语要点中英对照

线性代数 英文讲义

线代名词中英文对照

(完整版)线性代数笔记

LINEAR ALGEBRA 线性代数 课文 翻译

线性代数_常用单词词组_中英对照

山东大学《线性代数》课件01-5矩阵的初等变换与矩阵的秩

线性代数英文课件：ch1_1 Definition

线性代数-矩阵的初等变换

线代英文词汇

(完整word版)线性代数课程专业词汇表

线性代数

线性代数术语中英文对照_第一章

高等数学【线性代数】英文版课件1

线性代数英文讲义

线性代数课件第一章

线性代数英文讲义

LINEAR ALGEBRA 线性代数课文翻译