《线性代数英文讲义》doc版.doc

Chapter 1-Section 1 Systems of Linear 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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 Definitions A linear equation (线性方程) in n unknowns（未知量） is A linear system of m equations in n unknowns isThis is called a mxn(read as m by n) system.A solution to an mxn system is an ordered n-tuple of numbers （n 元数组） 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 SystemsEach equation can be represented graphically as a line in the plane. The ordered pair will be a solution if and only if it lies on both lines.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 many elements)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 systems Two 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 are written;(2) Multiply through one equation of a system by a nonzero real number;(3) Add a multiple of one equation to another equation. (subtract a 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 systems If 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 of is nonzero. (k=1, 2, ,n)An example of a system in strict triangular form: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 the s. we will refer to this array as the coefficient matrix (系数矩阵) of the system.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 matrixWe 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. s are placeholders that can be omitted until the end of 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 pivot number is called a pivotal row（主行）. We interchange the rows (if necessary) so that the pivotal row is the new first row. Multiples of the pivotal row are then subtracted form each of the remaining n-1 rows so as to obtain 0s in the first entries 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 the pivot element is then interchanged with the second row ( if necessary) of the matrix and is used as the new pivotal row. Multiples of the pivotal row are then subtracted form each of the remaining n-2 rows so as to eliminate all entries below the 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）.Assignment Students should be able to do all problems.Hand-in problems are: # 7-#11Chapter 1-Section 2 Row Echelon form2. 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 system Homogeneous 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 matrix.(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). The equations represented by the last two rows are:Since there are no 5-tuples that could possibly satisfy these equations, the system is inconsistent. Change the system above to a consistent system. 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:which is strictly triangular in the unknown . Thus for each pair of values assigned to and , 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 the rows 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. Otherwise, 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 rowsthe number of unknowns), the system will have a unique solution. If the number of nonzero rowsthe 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 (mn). 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,Put the free variables on the right-hand side, it follows thatThus for any real numbers and , the 5-tupleis a solution.Thus all ordered 5-tuple of the form are solutions to the system.2.4 Reduced Row Echelon FormDefinition: 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 e