线性代数Linear讲解
Linear_algebra_1
Linear algebra
徐 明 华 Xu Minghua 数理学院 School of Mathematics and Physics, Changzhou University 2010年1月
前言
研究内容 线 性 代 数(Linear Algebra)是 数 学 的 一 个 分 支, 它 的 研 究 对 象 是 向 量 , 向 量 空 间 (或 称 线 性 空 间 ), 线 性 变 换 和 有 限 维 线 性 方 程 组 . 矩 阵 是 用 来 表示向量之间的关系、线性变换、线性方程组及其解的重要工具. 应用领域 • 在 数 学 、 力 学 、 物 理 、 管 理 等 学 科 中 有 各 种 重 要 应 用; 是 计 算 机 图 形 学、计算机辅助设计、密码学等学科的基础或工具; • 是 进 行 科 学 计 算 的 基 础; 实 际 问 题 大 多 数 为 非 线 性 问 题, 而 非 线 性 问 题 通 常 可 以 被 近 似 为 线 性 问 题, 即 可 以 线 性 化, 随 着 计 算 机 的 发 展, 线 性 问 题 又 可 以 计 算 出 来, 因 此, 线 性 代 数 是 解 决 这 些 实 际 问 题 给 定 的 四 个 数 : a11, a12, a21, a22, 按 照 下 述 方 式 排 成 队 二 行 二 列的数表 a11 a21 a11 a21 a12 a22 a12 a22,
规 定表 达式 a11a22 − a12a21为上 述 数表 的二 阶行 列 式, 记 为 = a11a22 − a12a21,
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• 当 a < b时 , 经对 换后a的逆 序 数增 加1, 而b的逆 序数 不 变; • 当 a > b时 , 经对 换后a的逆 序 数不 变, 而 b的 逆序 数减 少1. 所以下面两个排列 a1 · · · al abb1 · · · bm, 的奇偶性不同. 再证一般对换的情形. 设排列 a1 · · · al ab1 · · · bmbc1 · · · cn 交 换a, b得到 a1 · · · al bb1 · · · bmac1 · · · cn. (2) (1) a1 · · · al bab1 · · · bm
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Linear Algebra总结
MATRICES· SOME DEFINITIONS
• Matrix: A rectangular array of numbers (named with capital letters) called entries with the size of the matrix described by the number of rows (horizontals) and columns (verticals); for example, a 3 by 4 matrix (3 X 4) has 3 rows and 4 columns;
o ~ 1 -2lFra bibliotek7 300 5 and this is a diagonal matrix 0 - 2 0 0 0 1
• Identity matrix (denoted by I): A square matrix with entries that are all' zeros except entries on the main diagonal, which must all equal the number one • Triangular matrix: A square matrix with all entries below the main diagonal equal to zero (upper triangular), or with all entries above the main diagonal equal to zero (lower triangular) • Equal matrices: Are the same size and have equal entries • Zero matrix: Every entry is the number zero • Scalar: A magnitude or a multiple • Row equivalent matrices: Can be produced through a sequence of row operations, such as: • Row interchange: Interchanging any 2 rows • Row scaling: Multiplying a row by any nonzero number • Row addition: Replacing a row with the sum of itself and any other row or multiple of that other row • Column equivalent matrices: Can be produced through a sequence of column operations, such as: • Column interchange: interchanging any 2 columns • Column scaling: multiplying a column by any nonzero number • Column addition: replacing a column with the sum of itself and any other column or multiple of that other column. • Elementary matrices: Square matrices that can be obtained from an identity matrix, I, of the same dimensions through a single row operation • The rank of matrix A, denoted rank(A), is the common dimension of the row space and column space of matrix A • The nullity of matrix A, denoted nullity(A), is the dimension of the nullspace of A
线性代数要点讲解
这组基向量之间是线性无关的，线性无关也就是说空间的张成需要刚刚好不多不少的向量张成，没有产生多余的向量
该组向量的任意线性组合形成的空间的维度恰好和基向量的个数是一致的
linear transformation，线性函数，input vectors 经过线性变换output vectors
1/22
线性组合
在三维空间里面找一个含有某个向量的子空间===》scalar*该向量=span（v）
在三维空间里面找一个含有二个具体向量的子空间===》scalar*向量+scalar*向量2=span（u，v）
也就是 span（V）=linear combination
在一个vector space中，有若干个vectors，这几个vectors的所有线性组合叫做这几个vectors的span
1/23
由于线性变换会涉及区域面积大小的变化===》自然就会问，变化的大小，前后的比例是多少呢？
===》我们给这个比例一个标签就是行列式，线性变换`矩阵，也就矩阵的行列式
===》然而行列式的值可以是negtive也就意味着图像进行了orientation变换
在二维空间里的orienta，我们知道变换有很多种，比如旋转，翻面等等
变换是input一个object，变换后，output a new object
因此，你也可以用function代替transformation，但是transformation更具有动态性的意义在里面
现在，我们仅研究简单的一种变换，叫做linear transformation
线性代数
系数行列式
二阶行列式. 二阶行列式.
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二. 三阶行列式 类似地, 类似地 为讨论三元线性方程组
a11 x 1 + a 12 x 2 + a13 x 3 = b1 a 21 x 1 + a 22 x 2 + a 23 x 3 = b2 a x + a x + a x = b 32 2 33 3 3 31 1
经 济 数 学 基 础
1
课程的作用
线性代数（ 线性代数（Linear Algebra）是代数学的一个分 这一词在我国出现较晚， 支，“Algebra”这一词在我国出现较晚，清代著名的数 学家、翻译家李善兰将它翻译成代数学，一直沿用至今。 学家、翻译家李善兰将它翻译成代数学，一直沿用至今。 线性代数是一门非常重要的基础课。 线性代数是一门非常重要的基础课。线性代数主要 处理线性关系的问题，其含义不断扩大， 处理线性关系的问题，其含义不断扩大，它的理论不仅 渗透到了数学的许多分支中，而且还在国民经济、工程 渗透到了数学的许多分支中，而且还在国民经济、 技术、理论物理、理论化学、航天、 技术、理论物理、理论化学、航天、航海等领域中都有 广泛的应用。 广泛的应用。 该课程对于培养学生的逻辑推理和抽象思维能力， 该课程对于培养学生的逻辑推理和抽象思维能力，空 间想象能力具有重要作用。通过线性代数的学习， 间想象能力具有重要作用。通过线性代数的学习，能使 学生获得应用学科中常用的矩阵、线性方程组等理论， 学生获得应用学科中常用的矩阵、线性方程组等理论， 具有熟练的矩阵运算能力和用矩阵方法解决实际问题的 能力。 能力。
a11 D = a21 a31 a12 a22 a32 a13 a23 a33
记
a11 b1 a13 D2 = a21 b2 a23 a31 b3 a33
线性代数(第五版)课件
• 想搞软件工程，好，3D游戏的数学基础就 是以图形的矩阵运算为基础；当然，如果 你只想玩3D游戏可以不必掌握线代；想搞 图像处理，大量的图像数据处理更离不开 矩阵这个强大的工具，《阿凡达》中大量 的后期电脑制作没有线代的数学工具简直 难以想象。
• 想搞经济研究。好，知道列昂惕夫（Wassily Leontief）吗？哈佛大学教授，1949年用计 算机计算出了由美国统计局的25万条经济数 据所组成的42个未知数的42个方程的方程组， 他打开了研究经济数学模型的新时代的大门。
这些模型通常都是线性的，也就是说，它们
是用线性方程组来描述的，被称为列昂惕夫 “投入-产出”模型。列昂惕夫因此获得了 1973年的诺贝尔经济学奖。
• 相当领导，好，要会运筹学，运筹学的一 个重要议题是线性规划。许多重要的管理 决策是在线性规划模型的基础上做出的。 线性规划的知识就是线代的知识啊。比如， 航空运输业就使用线性规划来调度航班， 监视飞行及机场的维护运作等；又如，你 作为一个大商场的老板，线性规划可以帮 助你合理的安排各种商品的进货，以达到 最大利润。
§1 二阶与三阶行列式
我们从最简单的二元线性方程组出发，探 求其求解公式，并设法化简此公式.
一、二元线性方程组与二阶行列式
二元线性方程组
aa1211
x1 x1
a12 x2 a22 x2
b1 b2
由消元法，得
(a11a22 a a 12 21 ) x1 b1a22 a12b2
(a11a22 a a 12 21 ) x2 a11b2 b1a21
二、线性代数的课程特点
高度的抽象性和严密逻辑性,并缺乏直观 的思维模型.
开设时间为大一、大二年级. 线性代数课时短, 内容多. 理论多, 例题少.
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—Linear Algebra
λ1 =1
λ2 =1/2
其中λ1, λ2 是矩陣A的特徵值，我們令det(A-λI)=0求出λ1, 與λ2，而特徵向量x1屬於A-I的nullspace ，而特徵向量x2屬
所有對應同樣特徵值的特徵向量形成一個子空間。
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6.1 Introduction to Eigenvalues : Good News, Bad News
Elimination does not preserve theλ’s .
1 A= 1 1 1 λ= 0 and λ= 2 1 U = 0 1 0
When a matrix is shifted by I, each λ is shifted by 1. 但是特徵向量不變， x1=(1, 1) 與 x2 =(1, -1)
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6.1 Introduction to Eigenvalues
範例：排序矩陣的特徵值λ滿足|λ|=1
上頁映射矩陣R也是排序矩陣(列對調)，其特徵值λ為1,-1 0 1 R= 1 0 但是P4的特徵值λ為±1, ± i有複數，其對應之特徵向量 (1, ±1, 1, ±1)與 (1,ction to Eigenvalues
範例：映射矩陣R的特徵值λ為1,-1
0 R= 1 1 0
(R2的特徵值是λ2=1 ,特徵向量不變)
R=2P- I , R2 = I
if Px= λx then 2Px=2λx , (2P- I) x=(2λ-1)x=Rx
0 1 P4 = 0 0
2011/6/18
0 0 1 0 0 0 1 0 0 0 1 0
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第七 章 Linear Transformations (線性轉 換)
7.1 7.2 7.3 7.4
The idea of a linear Transformation The Matrix of a Linear Transformation Change of Basis Diagonalization and the Pseudoinverse
linearity: T(cv+dw) = c T(v) +d T(w)
依據”linear”的性質 T(0)=0
例子：矩陣乘法運算T(v)=Av就是一個線性轉換 例子：T(v)=v +u0 不是一個線性轉換，除非u0=0 T(v)=v 稱為“identity transformation ”
例題3：T(v)=旋轉v向量30o (xy平面) 是一個線性轉換。
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7.1 The idea of a linear Transformation: Lines to Lines, Triangles to Triangles
下圖說明input線上的點對應到output線上的點，其間 保持等距關係。
The rule of linearity extends to combinations of three vectors or n vectorsu. c1v1 c2v2 ... cnvn T (u) c1T (v1 ) c2T (v2 ) ... cnT (vn ) 例題4：T(v)=投影v向量(in R3) 到 xy平面。
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7.1 The idea of a linear Transformation
例題5：T(v)=投影v向量(in R3) 到 z=1平面。
解： Let v=(v1,v2,v3 ) , then T(v)=(v1,v2,1) , T(cv)=(cv1, cv2,1) ≠cv hence the transformation is not linear.
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7.1 The idea of a linear Transformation: Linear Transformations of the plane
假設平面上一間”房子”有11個頂點 vi=(xi,yi), i=1,…,11. 我們做一個線性轉換，將這11個頂點對應到 頂點，而且他們之間的直線對應到直線，來產生新的” 房子”。 觀察不同的矩陣所產生的效果 6 3. 3 0 0 6 6 6 7 0 7 6
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7.1 The idea of a linear Transformation
定義： A transformation T assigns an output T(v) in output space W to each input vector v in input space V . The transformation is linear if it meets these requirements for all v and w : (a) T(v+w) = T(v) +T(w) (b) T(cv)=c T(v) for all c in R
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7.1 The idea of a linear Transformation
例子：T(v)=Av +u0 ”linear-plus-shift transformation” 不是一個線性轉換 例題1：假設a =(1,3,4), T(v)= a ·v (inner product) 是一個線性轉換。
1 1 1 1 u v w T ( u) T ( v ) T ( w ) 2 2 2 2
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7.1 The idea of a linear Transformation
註：”transformation” 有自己定義使用的語言，雖然可能沒 有用到矩陣，但相同的觀念依然可使用。 Range of T =set of all outputs T(v) : corresponds to column space Kernel of T =set of all inputs for which T(v)=0 : corresponds to nullspace The range is in the output space W The kernel is in the input space V
解：令 A = [1 3 4]，則 T(v)= a ·v = A v
解：因為 ||v+ w||≤ ||v||+ ||w|| ，而且 T(-v)= ||-v|| = ||-v||≠- ||v|| 解：我們將整個平面旋轉30o，會使得這個轉換滿足線性關係 。在這裡不用提及矩陣。
例題2：The length T(v)= ||v|| 不是一個線性轉換。
例題6：假設A是可逆矩陣，而且T(v)= Av則存 在逆線性轉換(inverse transform) T-1， 使得T-1(T(v))= v。
解： Let T-1(w )=A-1w, for all w in the range of T then T-1 is linear and T-1(T(v))= T-1(w) =A-1w =A-1Av=v , for w =T(v)=Av
解： The range is the xy plane , the kernel is the z-axis. Let v=(v1,v2,v3 ) , then T(v)=(v1,v2,0)=0 , hence v1 =0=v2 , the transformation (projection) is linear.