【正文】 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。(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。(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 n2 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 n1.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 n1 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（梯形矩陣）.Assignment Students should be able to do all problems.Handin problems are: 711167?！?.(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(梯形矩陣). The equations represented by the last two rows are:Since there are no 5tuples that could possibly satisfy these equations, the system is inconsistent. Change the system above to a consistent system. 224。Put the free variables on the righthand side, it follows thatThus for any real numbers and , the 5tupleis a solution.Thus all ordered 5tuple of the form are solutions to the system. 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 GaussianJordan reduction.The procedure for solving a linear system:(i) Write down the augmented matrix associated to the system。(iii) If the system if consistent, reduce the row echelon form into a reduced row echelon form.(iv) Write the solution in an ntuple formRemark: Ma