【正文】 Chapter 1 Matrices and Systems of EquationsLinear systems arise in applications to such areas as engineering, physics, electronics, business, economics, sociology（社會學）, ecology（生態(tài)學）, demography(人口統(tǒng)計學), and genetics（遺傳學）, etc.167。1. Systems of Linear EquationsNew words and phrases in this section:Linear equation 線性方程Linear system，System of linear equations 線性方程組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 等價的Equivalence 等價性Equivalent system 等價方程組Strict triangular system 嚴格上三角方程組Strict triangular form 嚴格上三角形式Back Substitution 回代法Matrix 矩陣Coefficient matrix 系數(shù)矩陣Augmented matrix 增廣矩陣Pivot element 主元Pivotal row 主行Echelon form 階梯形 Definitions A linear equation (線性方程) 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 ntuple of numbers （n 元數(shù)組） 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. 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. Equivalent systems Two systems of equations involving the same variables are said to be equivalent （等價的，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. 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 (嚴格三角形) if in the kth equation the coefficients of the first k1 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 (系數(shù)矩陣) of the system.A matrix （矩陣） is a rectangular array of numbersIf we attach to the coefficient matrix an additional column whose entries are the numbers on the righthand side of the system, we obtain the new matrixWe refer to this new matrix as the augmented matrix（增廣矩陣） 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 putation.Corresponding to the three operations used to obtain equivalent systems, the following row operation may be applied to the augmented matrix. 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 n1 rows so as to obtain 0’s 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