线性代数英文课件4.3
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 英文讲义
2 Matrix Representations of Linear Transformation
Theorem 4.2.1 If L is a linear transformation mapping Rn
into Rm, there is an m×n matrix A such that L(x)=Ax for each x Rn. In fact, the jth column vector of A is given x∈R by aj=L(ej) j=1, 2, …, n
Theorem 4.1.1 If L: V →W is a linear transformation
and S is a subspace of V, then (1) Ker(L) is a subspace of V. (2) L(S) is a subspace of W.
Example
Let L be the linear operator on R2 defined by
If there is another basis for
R2:
1 1 u1 = ， u 2 = 1 1
2 0 1 2 L(u1)=Au1= 1 1 1 = 2
2 0 1 2 L(u2)=Au2= 1 1 1 = 2
A is the matrix representing L relative to the ordered bases
If A is the matrix representing L with respect to the bases E and F and x=[v]E (the coordinate vector of v with respect to E) y=[w]F (the coordinate vector of w with respect to F) then L maps v into w if and only if A maps x into y.
线性代数(含全部课后题详细答案)4-3PPT课件
目
CONTENCT
录
• 课程介绍与教学目标 • 向量空间与线性变换 • 行列式与矩阵运算 • 特征值与特征向量 • 课后习题详解 • 课程总结与拓展延伸
01
课程介绍与教学目标
线性代数课程简介
线性代数是数学的一个分支， 研究线性方程组、向量空间、 矩阵等概念和性质。
简要介绍数值计算中常用的迭代法、插值 法、逼近法等基本方法，培养学生运用计 算机解决实际问题的能力。
简要介绍数学建模的基本思想和方法，通 过实例展示数学建模在解决实际问题中的 应用和价值。
THANK YOU
感谢聆听
05
课后习题详解
习题类型及解题思路
计算题
主要针对线性代数中的基本运算，如矩阵的加减、数乘和乘法等。解题思路通常是按照运算规则逐步进行，注意保持 矩阵的维度一致。
证明题
主要考察学生对线性代数基本定理和性质的理解和掌握。解题思路一般是从已知条件出发，结合相关定理和性质进行 推导，最终得出结论。
应用题
行列式性质
行列式具有线性性、交换性、倍加性 等基本性质，这些性质在行列式的计 算和证明中起到重要作用。
矩阵运算规则
矩阵加法
两个矩阵相加，要求它们具有相同的行数和列数， 对应元素相加。
矩阵数乘
一个数与矩阵相乘，将该数与矩阵中的每一个元素 相乘。
矩阵乘法
两个矩阵相乘，要求第一个矩阵的列数等于第二个 矩阵的行数，结果矩阵的行数等于第一个矩阵的行 数，列数等于第二个矩阵的列数。
将线性代数的知识应用于实际问题中，如求解线性方程组、矩阵的特征值和特征向量等。解题思路是首 先建立数学模型，将实际问题转化为线性代数问题，然后利用相关知识进行求解。
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
1 0
0
1
0
2
?
0 0 1 1
0
1
0 0
0
0
0
0
0
0
0 3
0
2
?
1 1
2
0
1 0
0
1
0 0
0
0
0 3
0
2
?
1 1
0
1Leabharlann 1 0 0 301
0
2
0 0 0 0
√
0
0
0
0
1 0 0 3
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ppt3
Examples
(1) V={ …, -3, -1, 1, 3, 5, 7, …} V is not closed under addition because 1+3=4 V.
(2)
Z={ …, -2, -1, 0, 1, 2, 3, 4, …}
Z is closed under addition because for any a, b Z, a + b Z. Z is not closed under scalar multiplication because ½ is a scalar, for any odd a Z, (½)a Z.
Ch04_9
The Complex Vector Space Cn
Let (u1 , ..., un ) be a sequence of n complexnumbers. T heset of all such sequences is denotedC n . Let operat ions of addition and scalar multiplica tion (by a complexscalar c) be defined on C n as follows:
[ f (- f )](x) f ( x) (- f )(x) f ( x) - [ f ( x)] 0 0( x)
Thus [f + (-f )] = 0, -f is the negative of f.
V={ f | f(x)=ax2+bx+c for some a,b,c R}
Ch04_11
Subspaces
In general, a subset of a vector space may or may not satisfy the closure axioms. However, any subset that is closed under both of these operations satisfies all the other vector space properties.
线性代数英文课件：ch4_Review
x1
x3
1 3
x5 ;
x2
x3
2 3
x5 ;
x3
x3 ;
x4
5 3
x5 ;
x5
x5
1 0 1 0 1/3
0
1
1
0
2
/
3
0 0 0 1 5/3
0
0
0
0
0
x1 x2 x3 x4 x5
c1
1
1
1
0
0
c2
0
1 3 2 3
which means
(k k1 knr ) * k11 kn-r n-r 0 (2)
From question(1) we know
*,1, ,n-r are linearly independent,
(k k1 knr ) 0; k1 0, , kn-r 0. k k1 kn-r 0.
1 0 1 0 1/3
0
1
1
0
2
/
3
0 0 0 1 5/3
0 0 0 0 0
x1 x3 1 / 3;
x2
x3
2 / 3;
x3
x3 ;
x4 5 / 3.
x1 1 1 / 3
x2 x3 x4
c
1
1
0
2 0 5
/ /
3
3
,
c
R.
(6)Let matrix A=(1 ,2 ,3 ,4 ,5 )，solve Ax 0.
1, ,n-r are linearly independent.
k1 knr 0,
So *,1, ,n-r are linearly independent.
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
线性代数课件4-3向量的内积和Schmidt正交化
向量内积的性质
非负性
$mathbf{u} cdot mathbf{v} geq 0$，当且仅当
$mathbf{u}$与$mathbf{v}$同 向或反向时取等号。
交换律
$mathbf{u} cdot mathbf{v} = mathbf{v} cdot mathbf{u}$。
线性代数课件4-3向量 的内积和schmidt正 交化
contents
目录
• 向量的内积 • Schmidt正交化 • 向量的模 • 向量的外积
01
向量的内积
向量内积的定义
向量内积的定义为两个向量之间的点乘，记作$mathbf{u} cdot mathbf{v}$，计 算公式为：$mathbf{u} cdot mathbf{v} = u_1v_1 + u_2v_2 + cdots + u_nv_n$，其中$mathbf{u} = (u_1, u_2, ldots, u_n)$和$mathbf{v} = (v_1, v_2, ldots, v_n)$。
1
正交化后的向量组是正交的，即任意两个不同向 量的点积为0。
2
正交化后的向量组是单位向量组，即每个向量的 模长为1。
3
正交化后的向量组是线性无关的，即不存在不全 为零的系数使得这些系数的线性组合等于零向量 。
Schmidt正交化的计算方法
首先，将非正交向量组进行单位化，使得每个 向量的模长为1。
然后，通过线性变换将每个向量与其余向量进 行正交化，使得任意两个不同向量的点积为0。
计算步骤
02
03
注意事项
首先计算各个分量，然后根据这 些分量构造向量c。
《线性代数》课件第4章
此时A的第j列元素恰为αj表示成β1, β2,…, βt的线性组合时的
系数.
证明：若向量组a1,a2,…,as可由β1, β2,…, βt线性表示，即每个ai
均可由β1, β2,…, βt线性表示，则有
α1 = a11β1 + a21β2 + + at1βt = (β1, β2,
, βt )⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝aaa12t111 ⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟,
我们有下面的定理： 定理 1.1 矩阵的秩数=行秩数=列秩数.
例1.3 设
α1 = (1, 2, 0,1)T , α2 = (0,1,1,1)T , α3 = (1, 3,1, 2)T , α4 = (1,1,−1, 0)T
求此向量组的秩数及一个极大无关组.
解 考虑向量组构成的矩阵
A=(α1,
α2,
我们有下面的命题：
命题1.
1. α1, α2,…, αs线性无关； 2.方程x1α1 + x2α2 + … + xxαs只有零解 3. 对于任意一组不全为零的数c1,c2,…,cs均有
c1α1 + c2α2 + + csαs ≠ 0, 4. 对于任意一组数c1,c2,…,cs, 若c1α1 + c2α2 +
定义1.4 两个可以互相表示的向量组称为等价向量组.
容易看出： 1. 向量组的等价是一个等价关系； 2. 等价向量组的秩数相同； 3. 任何向量组等价于其极大无关组; 4. 两个向量组等价当且仅当它们的极大无关组等价.
最后我们给出化简向量组的一种技巧 为此先给出一个定义
定义1.5 设α1, α2,…, αs和β1, β2,…, βs是两个向量组， 若对于任意一组数c1,c2,…,cs均有
线性代数英文课件4.1
Unit 10: Determinants
(text reference: Section 4.1)
c V. Olds 2010
128
Unit 10
10
Determin are a special class of matrices. We have already seen one instance of a concept which is defined only for square matrices — the inverse matrix. That is, only a square matrix may have an inverse. In this unit we will (begin to) learn about another concept which is defined only for square matrices — the determinant of a matrix.
Unit 10 d, then det A = ad − cb. That is, if A is a 2 × 2 matrix, then det A = a11 a22 − a21 a12 . Example 10.1. Find the determinants of the following matrices: 1 2 2 0 (a) A = [5] (b) B = (c) C = 3 4 1 3
The number which is the determinant of a square matrix measures a certain characteristic of the matrix. In a more advanced study of matrix algebra, this characteristic is used for various purposes. In this course, the only way in which we will use this number is in its connection to the existence of the inverse of the matrix, and through that it’s application to SLE’s in which the coefficient matrix is a square matrix. For these purposes, what will matter to us is whether or not this number, the determinant of the matrix, is 0. But of course, in order to determine whether or not the determinant of a particular matrix is 0, we need to know how to calculate that number. Calculating the determinant of a square matrix is somewhat complicated. The definition is recursive, meaning that the calculation is defined in a straightforward way for small matrices, and then for larger matrices, the determinant is defined as being a calculation involving the determinants of smaller matrices, which are certain submatrices of the matrix. We could express this recursive definition of the determinant of a square matrix of order n as applying for all n ≥ 2, specifically defining only the determinant of a square matrix of order 1, i.e. a (square) matrix containing only a single number. However, the calculation for a 2 × 2 matrix is very straightforward — easier to think of as a special definition all on its own — so instead we use specific definitions for n = 1 and n = 2, and then define the determinant of a square matrix of order n > 2 in terms of determinants of submatrices of order n − 1, which are found by expressing them in terms of determinants of successively smaller submatrices until we get down to submatrices of order 2. The calculation of det A as defined in this way, when A is a square matrix of order n > 2, is not really as complicated as it will look. It’s just a matter of applying a certain formula carefully, as many times as necessary until we have expressed det A in terms of the determinants of 2 × 2 matrices. Those determinants are easy to find. So we start by defining det A for square matrices of order 1 and of order 2. When A is a 1 × 1 matrix, i.e. a matrix containing only one number, finding the particular number det A which is associated with that matrix is trivial. That number is the only number around — the single number that’s in the matrix. For a square matrix of order 2, i.e. a matrix containing 4 numbers arranged in a square, we have to do a little more work. But it’s a simple calculation. In fact, we can think of the calculation as “down products minus up products”, which is something we have seen before. But this time there’s only one down product, and only one up product, so it’s actually just “down product minus up product”.
线性代数英文课件4.2
144
Unit 11
Theorem 11.1. If matrix B is obtained from square matrix A by multiplying one row or column of A by some non-zero scalar c, then det B = c(det A).
Math 1229A/B
Unit 11: Properties of Determinants
(text reference: Section 4.2)
c V. Olds 2010
Unit 11
143
11
Properties of Determinants
In this section, we learn more about determinants. First, we observe some properties of determinants that allow us to calculate determinants more easily. We examine the effects on the determinant when the various kinds of elementary row operations are performed, so that we can easily see how the determinants of the various row-equivalent matrices are related to one another as we perform these operations. This allows us to calculate the determinant of a matrix by row-reducing the matrix (a procedure we already know well) to obtain a matrix whose determinant is easily calculated using facts we’ve already learnt in the previous section. We also learn some useful properties which allow us to calculate the determinant of a matrix from the determinants of one or more other matrices whose determinants we may already know. And finally we examine the relationship between determinants and inverses, which allows us to relate determinants to systems of linear equations, using what we already know about the implications of the existence of the inverse of a matrix for the number of solutions to the SLE which has that matrix as its coefficient matrix. Throughout all of this, of course, it is important to remember that we are only dealing with square matrices when we talk about determinants. That is, it is only for a square matrix that the characteristic “the determinant of the matrix” is defined. First, let’s think about what effect multiplying some row of a matrix by a non-zero scalar will have on the determinant. That is, let’s think about the relationship between det A and det B if matrix B is identical to matrix A except that one of the rows in B is the corresponding row of A multiplied by some c = 0. So suppose we have some n × n matrix A = [aij ]. Let B = [bij ] be the matrix obtained by multiplying one row, row k , by some non-zero scalar c. Then we know that bkj = cakj and bij = aij for all i = k . We can calculate det B by expanding along row k . Notice that when we form submatrices of B by deleting row k (and also some column of B ), the one row that’s different than in matrix A is deleted, so that in the submatrix of B obtained, each entry is just the corresponding entry from matrix A and therefore the entire submatrix of B is simply the corresponding submatrix of A. That is, we have Bkj = Akj . So when we expand along row k we get:
高等数学【线性代数】英文版课件1
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.
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
x1 x1 x1
+ −
x2 x2
− x3 + x3 − 2x3
= = =
6 2 0
1 1 −1 1 and we found the matrices with coefficient matrix A = 1 −1 1 0 −2 6 1 −1 1 6 −1 1 1 1 A(1) = 2 −1 A(2) = 1 2 A(3) = 1 0 0 −2 1 0 −2 1
Cramer’s Rule Consider a SLE Ax = b in which A is a square matrix of order n with det A = 0. We can form n new n × n matrices by replacing different columns of A by the column vector b. And if we do so, then we can directly find the value of xj in the unique solution to Ax = b using the determinant of one of these new matrices and the determinant of A. A fellow named Cramer developed a rule for doing this. Before we get to the rule, though, we need to define these new matrices and the notation we use to refer to them. Definition: Let Ax = b be any SLE in which A is a square matrix. We define the matrix A(j ) to be the matrix obtained by replacing column j of A with the column vector b.
159
−4 2 −6 −1
−0+0
(−4)(−1) − (−6)(2) = 4 − (−12) = 16
Finally, for A(3) we can expand along row 3 again: 1 1 6 1 6 det A(3) = det 1 −1 2 = 1 det − 0 + 0 = (1)(2) − (−1)(6) = 2 − (−6) = 8 −1 2 1 0 0 So using Cramer’s Rule, we see that the values of x1 , x2 and x3 in the unique solution to the system are: x1 = 16 det A(1) = = 4, det A 4 x2 = det A(2) 16 = = 4, det A 4 x3 = det A(3) 8 = =2 det A 4
1 −1 −1 1 0 −2
6 b= 2 0 instance columns 6 2 0
We form the matrix A(j ) by replacing the j th column of A by the column vector b. So for to form A(1) we write the numbers from b instead of the first column of A, and then write 2 and 3 of A as usual. And so forth. We get: 6 1 −1 1 6 −1 1 1 1 1 A(1) = 2 −1 A(2) = 1 2 A(3) = 1 −1 0 0 −2 1 0 −2 1 0
Unit 12 For A(2), it will be easiest to 1 det A(2) = det 1 1 = zero out column 1 again: 6 −1 1 6 −1 2 1 = det 0 −4 2 = 1 det 0 −2 0 −6 −1
This means that if these determinants are reasonably easy to find, using Cramer’s Rule can be an easier way to find the solution to a SLE than row reducing. (However if the determinants require a lot of work to calculate, then using Cramer’s Rule involves more work than row reducing. So that’s just obnoxious.) For instance determinants of 2 × 2 matrices are always easy to find, so Cramer’s Rule is a reasonably good way to solve a system of 2 equations in 2 unknowns, as long as the coefficient matrix is nonsingular. And if there are 0’s around then sometimes the determinants of larger square matrices are reasonably easy to find. The following examples show how Cramer’s Rule can be used to find the unique solution to a SLE with a nonsingular square coefficient matrix. It’s important to remember, though, that Cramer’s Rule simply doesn’t apply to a system whose coefficient matrix isn’t square, or has determinant 0. Example 12.2. Use Cramer’s Rule to find the unique solution to the SLE in Example 12.1. Solution: We have the SLE:
Theorem 12.1. Cramer’s Rule Let A be any square matrix of order n with det A = 0 and let b be any n × 1 column vector. Then in the unique solution to the system Ax = b, the value of the j th unknown, xj , is given by: xj = det A(j ) det A
Math 1229A/B
Unit 12: Applications of the Determinant
(text reference: Section 4.3)
c V. Olds 2010
Unit 12
157
12
Applications of the Determinant
We shall finish up the course by looking at a couple of other ways that the determinant of a square matrix can be used, that is, a couple of applications of the determinant of a square matrix. The first is a method of finding the solution to the SLE Ax = b when A is a nonsingular (i.e. invertible) square matrix, without row reducing or finding the inverse matrix. Instead, we calculate det A as well as the determinant of certain other matrices obtained from A and b. After that, we will learn how to find the inverse of square matrix A, when it exists, using det A and another matrix which is obtained using the cofactors of A. Again, this gives us another method for doing something which previously we could only do by row reducing.
Now, how do we use these matrices? Recall that as long as det A = 0, the SLE Ax = b has a unique solution. According to Mr. Cramer, the values of the xj ’s in the unique solution to such
1 6 −1 2 0 0
First, we find det A. We can zero out column 1 and then expand along that column: 1 1 −1 1 1 −1 −2 2 1 = det 0 −2 2 = 1 det det A = det 1 −1 −0+0 −1 −1 1 0 −2 0 −1 −1 = (−2)(−1) − (−1)(2) = 2 − (−2) = 4