高等代数教学辅导

\begin{equation*} \det A=\begin{vmatrix} 1&a&a^2\\ 1&b&b^2\\ 1&c&c^2 \end{vmatrix}=(b-a)(c-a)(c-b)\neq 0, \end{equation*}

\begin{equation*} \det D_1=\begin{vmatrix} a^3&a&a^2\\ b^3&b&b^2\\ c^3&c&c^2 \end{vmatrix}=abc\begin{vmatrix} a^2&1&a\\ b^2&1&b\\ c^2&1&c \end{vmatrix}=abc(b-a)(c-a)(c-b), \end{equation*}

\begin{equation*} \det D_2=\begin{vmatrix} 1&a^3&a^2\\ 1&b^3&b^2\\ 1&c^3&c^2 \end{vmatrix}=-(b-a)(c-a)(c-b)(ab+ac+bc), \end{equation*}

\begin{equation*} \det D_3=\begin{vmatrix} 1&a&a^3\\ 1&b&b^3\\ 1&c&c^3 \end{vmatrix}=(b-a)(c-a)(c-b)(a+b+c), \end{equation*}

\begin{equation*} \left\{\begin{array}{ccl} x_1&=&\frac{\det D_1}{\det A}=abc,\\ x_2&=&\frac{\det D_2}{\det A}=-(ab+ac+bc),\\ x_3&=&\frac{\det D_3}{\det A}=a+b+c. \end{array}\right. \end{equation*}

\begin{equation*} \det A=\begin{vmatrix} a&-2&-1\\ 2&1&1\\ 10&5&4 \end{vmatrix}=-a-4. \end{equation*}

\begin{equation*} f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0(a_n\neq 0), \end{equation*}

\begin{equation} \left\{\begin{array}{c} f(c_1)=a_0+a_1c_1+a_2c_1^2+\cdots+a_nc_1^n=0,\\ f(c_2)=a_0+a_1c_2+a_2c_2^2+\cdots+a_nc_2^n=0,\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ f(c_{n+1})=a_0+a_1c_{n+1}+a_2c_{n+1}^2+\cdots+a_nc_{n+1}^n=0.\\ \end{array}\right.\tag{3.4.1} \end{equation}

\begin{equation*} \begin{vmatrix} 1&c_1&c_1^2&\cdots&c_1^n\\ 1&c_2&c_2^2&\cdots&c_2^n\\ \vdots&\vdots&\vdots&&\vdots\\ 1&c_{n+1}&c_{n+1}^2&\cdots&c_{n+1}^n \end{vmatrix}=\prod\limits_{1\leq i< j\leq n+1}(c_j-c_i)\neq 0, \end{equation*}

\begin{equation*} g_k(x)=\frac{(x-a_1)\cdots(x-a_{k-1})(x-a_{k+1})\cdots (x-a_n)}{(a_k-a_1)\cdots(a_k-a_{k-1})(a_k-a_{k+1})\cdots (a_k-a_n)}, \end{equation*}

\begin{equation*} L(x)=\sum\limits_{k=1}^n b_kg_k(x) \end{equation*}

\begin{equation*} g_k(a_i)=\left\{\begin{array}{ll} 1, & k=i,\\ 0, & k\neq i, \end{array}\right. \end{equation*}

\begin{equation*} L(a_i)=\sum\limits_{k=1}^n b_kg_k(a_i)=b_i. \end{equation*}

\begin{equation*} \left\{ \begin{array}{c} x_0+a_1x_1+a_1^2x_2+\cdots+a_1^{n-1}x_{n-1}=b_1,\\ x_0+a_2x_1+a_2^2x_2+\cdots+a_2^{n-1}x_{n-1}=b_2,\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ x_0+a_nx_1+a_n^2x_2+\cdots+a_n^{n-1}x_{n-1}=b_n, \end{array}\right. \end{equation*}

\begin{equation*} \det A=\begin{vmatrix} 1&a_1&a_1^2&\cdots&a_1^{n-1}\\ 1&a_2&a_2^2&\cdots&a_2^{n-1}\\ \vdots&\vdots&\vdots&&\vdots\\ 1&a_n&a_n^2&\cdots&a_n^{n-1} \end{vmatrix}=\prod\limits_{1\leq k<l\leq n}(a_l-a_k)\neq 0, \end{equation*}

\begin{equation} \left\{ \begin{array}{c} x_0+a_1x_1+a_1^2x_2+\cdots+a_1^{n-1}x_{n-1}=b_1,\\ x_0+a_2x_1+a_2^2x_2+\cdots+a_2^{n-1}x_{n-1}=b_2,\\ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\\ x_0+a_nx_1+a_n^2x_2+\cdots+a_n^{n-1}x_{n-1}=b_n, \end{array}\right.\tag{3.4.2} \end{equation}

\begin{equation*} \det A=\begin{vmatrix} 1&a_1&a_1^2&\cdots&a_1^{n-1}\\ 1&a_2&a_2^2&\cdots&a_2^{n-1}\\ \vdots&\vdots&\vdots&&\vdots\\ 1&a_n&a_n^2&\cdots&a_n^{n-1} \end{vmatrix}=\prod\limits_{1\leq k<l\leq n}(a_l-a_k)\neq 0, \end{equation*}

\begin{equation*} x_{j}=\frac{\det D_j}{\det A},\ j=0,1,\ldots,n-1, \end{equation*}

\begin{equation*} \det D_j=\begin{vmatrix} 1&a_1&\cdots&a_1^{j-1}&b_1&a_1^{j+1}&\cdots&a_1^{n-1}\\ 1&a_2&\cdots&a_2^{j-1}&b_2&a_2^{j+1}&\cdots&a_2^{n-1}\\ \vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots\\ 1&a_n&\cdots&a_n^{j-1}&b_n&a_n^{j+1}&\cdots&a_n^{n-1} \end{vmatrix}. \end{equation*}

\begin{equation*} L(x)=c_0+c_1x+\cdots+c_{n-1}x^{n-1}, \end{equation*}

\begin{equation*} c_j=\frac{\det D_j}{\det A},\ j=0,1,\ldots,n-1, \end{equation*}

\begin{equation*} L(x)=\sum\limits_{i=1}^n \left(b_i\cdot\prod\limits_{\begin{array}{c}1\leq j\leq n\\j\neq i\end{array}}\frac{x-a_j}{a_i-a_j}\right). \end{equation*}

\begin{equation*} \det D_j=\sum\limits_{i=1}^n(-1)^{i+j+1}b_iM_{i,j+1}, \end{equation*}

\begin{equation*} M_{i,j+1}=\begin{vmatrix} 1&a_1&\cdots&a_1^{j-1}&a_1^{j+1}&\cdots&a_1^{n-1}\\ \vdots&\vdots&&\vdots&\vdots&&\vdots\\ 1&a_{i-1}&\cdots&a_{i-1}^{j-1}&a_{i-1}^{j+1}&\cdots&a_{i-1}^{n-1}\\ 1&a_{i+1}&\cdots&a_{i+1}^{j-1}&a_{i+1}^{j+1}&\cdots&a_{i+1}^{n-1}\\ \vdots&\vdots&&\vdots&\vdots&&\vdots\\ 1&a_n&\cdots&a_n^{j-1}&a_n^{j+1}&\cdots&a_n^{n-1} \end{vmatrix}, \end{equation*}

\begin{equation*} \begin{array}{ccl} c_j&=&\frac{\sum\limits_{i=1}^n(-1)^{i+j+1}b_i\left(\prod\limits_{\begin{array}{c} 1\leq k<l\leq n\\ k,l\neq i\end{array}}(a_l-a_k)\right)\cdot\left(\sum\limits_{\begin{array}{c}1\leq i_1< \cdots<i_{n-1-j}\leq n\\i_1,\ldots ,i_{n-1-j}\neq i\end{array}}a_{i_1}\cdots a_{i_{n-1-j}}\right)}{\prod\limits_{1\leq k<l\leq n}(a_l-a_k)}\\ &=&\sum\limits_{i=1}^n\left(\prod\limits_{\begin{array}{c} 1\leq k\leq n\\k\neq i\end{array}}\frac{b_i}{a_i-a_k}\right)\cdot\left((-1)^{n-j-1}\sum\limits_{\begin{array}{c}1\leq i_1< \cdots<i_{n-1-j}\leq n\\i_1,\ldots ,i_{n-1-j}\neq i\end{array}}a_{i_1}\cdots a_{i_{n-1-j}}\right). \end{array} \end{equation*}

\begin{equation*} (-1)^{n-j-1}\sum\limits_{\begin{array}{c}1\leq i_1< \cdots<i_{n-1-j}\leq n\\i_1,\ldots ,i_{n-1-j}\neq i\end{array}}a_{i_1}\cdots a_{i_{n-1-j}} \end{equation*}

\begin{equation*} \sum\limits_{i=1}^n \left(b_i\cdot\prod\limits_{\begin{array}{c} 1\leq k\leq n\\k\neq i\end{array}}\frac{x-a_k}{a_i-a_k}\right) \end{equation*}

\begin{equation*} L(x)=\sum\limits_{i=1}^n \left(b_i\cdot\prod\limits_{\begin{array}{c} 1\leq k\leq n\\k\neq i\end{array}}\frac{x-a_k}{a_i-a_k}\right). \end{equation*}

\begin{equation*} ax+by+c=0, \end{equation*}

\begin{equation*} \left\{\begin{array}{l} ax+by+c=0,\\ ax_1+by_1+c=0,\\ ax_2+by_2+c=0 \end{array}\right. \end{equation*}

\begin{equation*} \begin{vmatrix} x & y & 1\\ x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ \end{vmatrix} = 0. \end{equation*}