论文部分内容阅读
历年来在高等代数的教学中,总发現某些学生对方程有着模糊的概念。例如,按照現行教材,中学毕业生进入高等学校后第一次接触到方程概念的是克萊姆規則:n个未知量n个方程的綫性方程組 a_(11)x~1+a_(12)x_2+ …+a_(1n)x_n=b_1, a_(21)x_1+a_(22)x_2+…+a_(2n)x_n=b_2, a_(n1)+a_(n2)x_2+…+a_(nn)x_n=b_n (1)的系数行列式D=|aij≠0时,(1)有解且仅有一解,即x_i=Di/D,i=1,2,…,n。 証明分两步:第一步是假定(1)有解,得出xi=Di/D。第二步是用真x_i=Di/D代入(1),得出真的等式,因而x_i=Di/D的确是(1)的解。較多的同学感到第二步是多余的,沒有必要。另一个例子是在討論向量方程
In the teaching of advanced algebra over the years, it has been found that some students have a vague concept of equations. For example, according to the current teaching materials, the first time the middle school graduates came into contact with the concept of equations after entering a higher school is the Klem rule: a system of n equations with unknown variables n equations a_(11)x~1+a_(12 X_2+ ...+a_(1n)x_n=b_1, a_(21)x_1+a_(22)x_2+...+a_(2n)x_n=b_2, a_(n1)+a_(n2)x_2+...+a_(nn)x_n When the coefficient determinant of =b_n (1) is D=|aij ≠ 0, (1) has a solution and there is only one solution, that is, x_i=Di/D, i=1, 2,...,n. The proof is in two steps: The first step is to assume (1) that there is a solution and that xi = Di/D. The second step is to substitute (1) with true x_i=Di/D to get a true equation, so x_i=Di/D is indeed a solution of (1). More students felt that the second step was redundant and unnecessary. Another example is discussing vector equations