&p&语料库语言学基础知识写作计划：&/p&&ul&&li&矩阵（&b&Matlab&/b&, Python, R, &a href=&https://zhuanlan.zhihu.com/p/& class=&internal&&Haskell&/a&）&/li&&li&概率论（&a href=&https://zhuanlan.zhihu.com/p/& class=&internal&&基本概念、离散分布&/a&, 2）&/li&&li&统计学（Matlab, Python, R, Haskell）&/li&&li&机器学习（Matlab, Python, R, Haskell）&/li&&/ul&&p&&br&&/p&&blockquote&主要参考资料：Michael Artin 的 Algebra&/blockquote&&p&如果没有购买 Matlab，可以下载开源软件 Octave 和 Scilab，它们的语法和 Matlab 基本一致。&/p&&p&&br&&/p&&p&&b&一、基本概念&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&X&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span& &span class=&mi&&1&/span&&span class=&p&&,&/span& &span class=&mi&&0&/span&&span class=&p&&;&/span& &span class=&mi&&1&/span&&span class=&p&&,&/span& &span class=&mi&&3&/span&&span class=&p&&,&/span& &span class=&mi&&5&/span&&span class=&p&&]&/span& &span class=&c&&% 定义2*3 矩阵 matrix (复数 matrices)&/span&
&span class=&n&&X&/span& &span class=&p&&=&/span&
&span class=&mi&&2&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&3&/span&
&span class=&mi&&5&/span&
&span class=&nb&&size&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&)&/span& &span class=&c&&% 测量矩阵的行列数目（行 row，列 column）&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&span class=&n&&X&/span&&span class=&p&&(&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span& &span class=&c&&% 矩阵的项X_ij，其中i为行指数 row index，j为列指数 column index&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mi&&1&/span&
&span class=&n&&X&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mi&&3&/span&
&span class=&n&&A&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span& &span class=&c&&% 建立2*2的方块矩阵 square matrix&/span&
&span class=&n&&A&/span& &span class=&p&&=&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&n&&B&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&3&/span&&span class=&p&&)&/span& &span class=&c&&% 建立2*3的矩阵&/span&
&span class=&n&&B&/span& &span class=&p&&=&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&n&&R&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&3&/span&&span class=&p&&]&/span& &span class=&c&&% 行向量 row vector&/span&
&span class=&n&&R&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&span class=&n&&C&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&;&/span&&span class=&mi&&2&/span&&span class=&p&&;&/span&&span class=&mi&&3&/span&&span class=&p&&]&/span& &span class=&c&&% 列向量 column vector&/span&
&span class=&n&&C&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&二、基本运算&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&X&/span& &span class=&o&&+&/span& &span class=&n&&B&/span& &span class=&c&&% 矩阵的加法 addition （对应的元素相加，形状不同则无法相加）&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&2&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&3&/span&
&span class=&mi&&5&/span&
&span class=&mi&&2&/span& &span class=&o&&*&/span& &span class=&n&&X&/span& &span class=&c&&% 数乘 scalar multiplication&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&4&/span&
&span class=&mi&&2&/span&
&span class=&mi&&0&/span&
&span class=&mi&&2&/span&
&span class=&mi&&6&/span&
&span class=&mi&&10&/span&
&span class=&n&&Y&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&3&/span&&span class=&p&&;&/span&&span class=&mi&&4&/span&&span class=&p&&,&/span&&span class=&mi&&5&/span&&span class=&p&&,&/span&&span class=&mi&&6&/span&&span class=&p&&;&/span&&span class=&mi&&7&/span&&span class=&p&&,&/span&&span class=&mi&&8&/span&&span class=&p&&,&/span&&span class=&mi&&9&/span&&span class=&p&&];&/span&
&span class=&n&&Z&/span& &span class=&p&&=&/span& &span class=&n&&X&/span& &span class=&o&&*&/span& &span class=&n&&Y&/span& &span class=&c&&% 矩阵乘法 matrix multiplication，m*n 矩阵只能和 n*l矩阵相乘&/span&
&span class=&n&&Z&/span& &span class=&p&&=&/span&
&span class=&mi&&6&/span&
&span class=&mi&&9&/span&
&span class=&mi&&12&/span&
&span class=&mi&&48&/span&
&span class=&mi&&57&/span&
&span class=&mi&&66&/span&
&span class=&n&&X&/span& &span class=&o&&.*&/span& &span class=&n&&Z&/span& &span class=&c&&% 元素对应相乘 Hadamard product&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&12&/span&
&span class=&mi&&9&/span&
&span class=&mi&&0&/span&
&span class=&mi&&48&/span&
&span class=&mi&&171&/span&
&span class=&mi&&330&/span&
&/code&&/pre&&/div&&p&&i&线性方程的矩阵表示&br&&/i&&/p&&p&&img src=&https://www.zhihu.com/equation?tex=a_%7B11%7Dx_1+%2B+%5Ccdots+%2B+a_%7B1n%7Dx_n+%3D+b_1%5C%5C+a_%7B21%7Dx_1+%2B+%5Ccdots+%2B+a_%7B2n%7Dx_n+%3D+b_2%5C%5C+%5Cvdots+%5Chspace%7B2.2cm%7D%5Cvdots+%5Chspace%7B1cm%7D%5Cvdots%5C%5C+a_%7Bm1%7Dx_1+%2B+%5Ccdots+%2B+a_%7Bmn%7Dx_n+%3D+b_m%5C%5C& alt=&a_{11}x_1 + \cdots + a_{1n}x_n = b_1\\ a_{21}x_1 + \cdots + a_{2n}x_n = b_2\\ \vdots \hspace{2.2cm}\vdots \hspace{1cm}\vdots\\ a_{m1}x_1 + \cdots + a_{mn}x_n = b_m\\& eeimg=&1&&&/p&&p&可以写成 &img src=&https://www.zhihu.com/equation?tex=AX%3DB& alt=&AX=B& eeimg=&1&& ，其中 &img src=&https://www.zhihu.com/equation?tex=X& alt=&X& eeimg=&1&& 和 &img src=&https://www.zhihu.com/equation?tex=B& alt=&B& eeimg=&1&& 均为列向量。&/p&&p&&br&&/p&&p&&i&矩阵的分配率&/i&&/p&&p&&img src=&https://www.zhihu.com/equation?tex=A%28B%2BB%27%29%3DAB+%2B+AB%27%5C%5C& alt=&A(B+B')=AB + AB'\\& eeimg=&1&&&/p&&p&矩阵的乘法结合律&/p&&p&&img src=&https://www.zhihu.com/equation?tex=%28AB%29C%3DA%28BC%29%5C%5C& alt=&(AB)C=A(BC)\\& eeimg=&1&&&/p&&p&&i&标量位置可变性&/i&&/p&&p&&img src=&https://www.zhihu.com/equation?tex=c%28AB%29%3D%28cA%29B%3DA%28cB%29%5C%5C& alt=&c(AB)=(cA)B=A(cB)\\& eeimg=&1&&&/p&&p&&i&矩阵乘法不满足交换律&/i&&/p&&p&&img src=&https://www.zhihu.com/equation?tex=%5Cexists+A%2CB+%5BAB%5Cneq+BA%5D%5C%5C& alt=&\exists A,B [AB\neq BA]\\& eeimg=&1&&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&Zuo&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&;&/span&&span class=&mi&&0&/span&&span class=&p&&,&/span&&span class=&mi&&0&/span&&span class=&p&&];&/span& &span class=&n&&You&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&0&/span&&span class=&p&&;&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&];&/span&
&span class=&n&&Zuo&/span& &span class=&o&&*&/span& &span class=&n&&You&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&3&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&n&&You&/span& &span class=&o&&*&/span& &span class=&n&&Zuo&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&2&/span&
&span class=&mi&&2&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&c&&% 如果AB=BA，我们称A和B可交换 commute&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&三、一些特殊的矩阵&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&c&&% 对角线项 &/span&
&span class=&c&&% 对角矩阵diagonal matrix：所有非零项都是对角线项&/span&
&span class=&n&&I&/span& &span class=&p&&=&/span& &span class=&nb&&eye&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span&
&span class=&c&&% 单位矩阵 identity matrix：对角线项为1的对角矩阵&/span&
&span class=&n&&I&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&n&&Z&/span& &span class=&o&&*&/span& &span class=&nb&&eye&/span&&span class=&p&&(&/span&&span class=&mi&&3&/span&&span class=&p&&)&/span& &span class=&o&&==&/span& &span class=&n&&Z&/span& &span class=&c&&% 单位矩阵右可消除&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mi&&2&/span&×&span class=&mi&&3&/span& &span class=&n&&logical&/span& &span class=&n&&array&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&nb&&eye&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span& &span class=&o&&*&/span& &span class=&n&&Z&/span& &span class=&o&&==&/span& &span class=&n&&Z&/span& &span class=&c&&% 单位矩阵左可消除&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mi&&2&/span&×&span class=&mi&&3&/span& &span class=&n&&logical&/span& &span class=&n&&array&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&mi&&1&/span&
&span class=&nb&&triu&/span&&span class=&p&&(&/span&&span class=&n&&Y&/span&&span class=&p&&)&/span& &span class=&c&&% 获取矩阵的上三角 upper triangular&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&span class=&mi&&0&/span&
&span class=&mi&&5&/span&
&span class=&mi&&6&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&9&/span&
&span class=&nb&&tril&/span&&span class=&p&&(&/span&&span class=&n&&Y&/span&&span class=&p&&)&/span& &span class=&c&&% 获取矩阵的下三角 lower triangular&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&4&/span&
&span class=&mi&&5&/span&
&span class=&mi&&0&/span&
&span class=&mi&&7&/span&
&span class=&mi&&8&/span&
&span class=&mi&&9&/span&
&span class=&n&&D&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&;&/span&&span class=&mi&&5&/span&&span class=&p&&,&/span&&span class=&mi&&3&/span&&span class=&p&&];&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&四、逆矩阵&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&inv&/span&&span class=&p&&(&/span&&span class=&n&&Y&/span&&span class=&p&&)&/span& &span class=&c&&% 逆 inverse&/span&
&span class=&n&&Warning&/span&&span class=&p&&:&/span& &span class=&n&&Matrix&/span& &span class=&n&&is&/span& &span class=&n&&close&/span& &span class=&n&&to&/span& &span class=&n&&singular&/span& &span class=&n&&or&/span& &span class=&n&&badly&/span& &span class=&n&&scaled&/span&&span class=&p&&.&/span& &span class=&n&&Results&/span& &span class=&n&&may&/span& &span class=&n&&be&/span& &span class=&n&&inaccurate&/span&&span class=&p&&.&/span&
&span class=&n&&RCOND&/span& &span class=&p&&=&/span&
&span class=&mf&&2.&/span&&span class=&p&&.&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mf&&1.0e+16&/span& &span class=&o&&*&/span&
&span class=&mf&&0.3153&/span&
&span class=&o&&-&/span&&span class=&mf&&0.6305&/span&
&span class=&mf&&0.3153&/span&
&span class=&o&&-&/span&&span class=&mf&&0.6305&/span&
&span class=&mf&&1.2610&/span&
&span class=&o&&-&/span&&span class=&mf&&0.6305&/span&
&span class=&mf&&0.3153&/span&
&span class=&o&&-&/span&&span class=&mf&&0.6305&/span&
&span class=&mf&&0.3153&/span&
&span class=&n&&inv&/span&&span class=&p&&(&/span&&span class=&n&&D&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mf&&3.0000&/span&
&span class=&o&&-&/span&&span class=&mf&&1.0000&/span&
&span class=&o&&-&/span&&span class=&mf&&5.0000&/span&
&span class=&mf&&2.0000&/span&
&span class=&n&&inv&/span&&span class=&p&&(&/span&&span class=&n&&Y&/span&&span class=&p&&)&/span& &span class=&o&&*&/span& &span class=&n&&Y&/span&
&span class=&n&&Warning&/span&&span class=&p&&:&/span& &span class=&n&&Matrix&/span& &span class=&n&&is&/span& &span class=&n&&close&/span& &span class=&n&&to&/span& &span class=&n&&singular&/span& &span class=&n&&or&/span& &span class=&n&&badly&/span& &span class=&n&&scaled&/span&&span class=&p&&.&/span&
&span class=&n&&Results&/span& &span class=&n&&may&/span& &span class=&n&&be&/span& &span class=&n&&inaccurate&/span&&span class=&p&&.&/span& &span class=&n&&RCOND&/span& &span class=&p&&=&/span&
&span class=&mf&&2.&/span&&span class=&p&&.&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&o&&-&/span&&span class=&mi&&2&/span&
&span class=&mi&&0&/span&
&span class=&mi&&2&/span&
&span class=&mi&&4&/span&
&span class=&mi&&0&/span&
&span class=&mi&&12&/span&
&span class=&o&&-&/span&&span class=&mi&&2&/span&
&span class=&mi&&0&/span&
&span class=&o&&-&/span&&span class=&mi&&2&/span&
&span class=&n&&inv&/span&&span class=&p&&(&/span&&span class=&n&&D&/span&&span class=&p&&)&/span& &span class=&o&&*&/span& &span class=&n&&D&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&n&&pinv&/span&&span class=&p&&(&/span&&span class=&n&&Y&/span&&span class=&p&&)&/span& &span class=&c&&% 广义逆 Moore-Penrose Pseudoinverse &/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&o&&-&/span&&span class=&mf&&0.6389&/span&
&span class=&o&&-&/span&&span class=&mf&&0.1667&/span&
&span class=&mf&&0.3056&/span&
&span class=&o&&-&/span&&span class=&mf&&0.0556&/span&
&span class=&mf&&0.0000&/span&
&span class=&mf&&0.0556&/span&
&span class=&mf&&0.5278&/span&
&span class=&mf&&0.1667&/span&
&span class=&o&&-&/span&&span class=&mf&&0.1944&/span&
&span class=&n&&pinv&/span&&span class=&p&&(&/span&&span class=&n&&D&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mf&&3.0000&/span&
&span class=&o&&-&/span&&span class=&mf&&1.0000&/span&
&span class=&o&&-&/span&&span class=&mf&&5.0000&/span&
&span class=&mf&&2.0000&/span&
&span class=&c&&% inv(X) 返回错误&/span&
&span class=&n&&pinv&/span&&span class=&p&&(&/span&&span class=&n&&X&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mf&&0.4333&/span&
&span class=&o&&-&/span&&span class=&mf&&0.0333&/span&
&span class=&mf&&0.1333&/span&
&span class=&mf&&0.0667&/span&
&span class=&o&&-&/span&&span class=&mf&&0.1667&/span&
&span class=&mf&&0.1667&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&引理1（左右逆等同）&/b&&/p&&p&设 &img src=&https://www.zhihu.com/equation?tex=A& alt=&A& eeimg=&1&& 为方块矩阵，且有&b&右逆（right inverse）&/b& &img src=&https://www.zhihu.com/equation?tex=R& alt=&R& eeimg=&1&& ，使得 &img src=&https://www.zhihu.com/equation?tex=AR%3DI& alt=&AR=I& eeimg=&1&& ，并有&b&左逆（left inverse） &/b&&img src=&https://www.zhihu.com/equation?tex=L& alt=&L& eeimg=&1&& ，使得 &img src=&https://www.zhihu.com/equation?tex=LA%3DI& alt=&LA=I& eeimg=&1&& 。则 &img src=&https://www.zhihu.com/equation?tex=L%3DR& alt=&L=R& eeimg=&1&& 。故 &img src=&https://www.zhihu.com/equation?tex=A& alt=&A& eeimg=&1&& 可逆且 &img src=&https://www.zhihu.com/equation?tex=R& alt=&R& eeimg=&1&& 为其逆矩阵。&/p&&p&&br&&/p&&p&&b&命题2&/b&&/p&&p&设 &img src=&https://www.zhihu.com/equation?tex=A%2CB& alt=&A,B& eeimg=&1&& 为可逆 &img src=&https://www.zhihu.com/equation?tex=n%5Ctimes+n& alt=&n\times n& eeimg=&1&& 矩阵，则它们的积 &img src=&https://www.zhihu.com/equation?tex=AB& alt=&AB& eeimg=&1&& 亦可逆。 &img src=&https://www.zhihu.com/equation?tex=A& alt=&A& eeimg=&1&& 的逆矩阵 &img src=&https://www.zhihu.com/equation?tex=A%5E%7B-1%7D& alt=&A^{-1}& eeimg=&1&& 亦可逆。且 &img src=&https://www.zhihu.com/equation?tex=%28AB%29%5E%7B-1%7D%3DB%5E%7B-1%7DA%5E%7B-1%7D& alt=&(AB)^{-1}=B^{-1}A^{-1}& eeimg=&1&& ， &img src=&https://www.zhihu.com/equation?tex=%28A%5E%7B-1%7D%29%5E%7B-1%7D%3DA& alt=&(A^{-1})^{-1}=A& eeimg=&1&& 。&/p&&p&&br&&/p&&p&&b&五、行列式（I）&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&D&/span&&span class=&p&&)&/span& &span class=&c&&% 行列式 determinant &/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mf&&1.0000&/span&
&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&Y&/span&&span class=&p&&)&/span& &span class=&c&&% 行列式近乎0；如果行列式为0，则矩阵不可逆&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&o&&-&/span&&span class=&mf&&9.5162e-16&/span&
&span class=&c&&% n 次一般线性群（n-dimensional general linear group）是 n×n 可逆矩阵的集合，&/span&
&span class=&c&&%和与之一起的普通矩阵乘法运算&/span&
&span class=&c&&% 如果一个矩阵一整行或一整列均为0，那么它是不可逆的&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&六、矩阵单位&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&E&/span& &span class=&p&&=&/span& &span class=&nb&&zeros&/span&&span class=&p&&(&/span&&span class=&mi&&3&/span&&span class=&p&&,&/span&&span class=&mi&&4&/span&&span class=&p&&);&/span&
&span class=&n&&E&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&3&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&mi&&1&/span& &span class=&c&&% 矩阵单位matrix unit e_2,3，最简单的非零矩阵&/span&
&span class=&n&&E&/span& &span class=&p&&=&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&c&&% 基basis：矩阵单位的集合&/span&
&span class=&c&&%n次实数空间标准基standard basis：长度为n的列向量的矩阵单位的集合&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&七、初等矩阵&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&c&&% Elementary matrices 对应着 elementary row operations&/span&
&span class=&c&&% 操作一：将一行的数量积加到另一行上&/span&
&span class=&n&&M&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span& &span class=&mi&&2&/span& &span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span&&span class=&p&&;&/span&&span class=&mi&&2&/span& &span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span&&span class=&p&&;&/span&&span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span&&span class=&p&&;&/span&&span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span& &span class=&mi&&8&/span&&span class=&p&&;&/span&&span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span& &span class=&mi&&8&/span& &span class=&mi&&9&/span&&span class=&p&&];&/span&
&span class=&n&&E1&/span& &span class=&p&&=&/span& &span class=&nb&&eye&/span&&span class=&p&&(&/span& &span class=&mi&&5&/span& &span class=&p&&)&/span&
&span class=&c&&% 生成 5x5 单位矩阵&/span&
&span class=&n&&E1&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&n&&E1&/span&&span class=&p&&(&/span&&span class=&mi&&4&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&mf&&0.1&/span&
&span class=&c&&% set entry (4,2) to 0.1，形成Type i 初等矩阵&/span&
&span class=&c&&% Type i 初等矩阵是在单位矩阵的基础上再加上一个非零项&/span&
&span class=&n&&E1&/span& &span class=&p&&=&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&0.1000&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&n&&E1&/span& &span class=&o&&*&/span& &span class=&n&&M&/span&
&span class=&c&&% add to row 4, 0.1 times row 2&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mf&&1.0000&/span&
&span class=&mf&&2.0000&/span&
&span class=&mf&&3.0000&/span&
&span class=&mf&&4.0000&/span&
&span class=&mf&&5.0000&/span&
&span class=&mf&&2.0000&/span&
&span class=&mf&&3.0000&/span&
&span class=&mf&&4.0000&/span&
&span class=&mf&&5.0000&/span&
&span class=&mf&&6.0000&/span&
&span class=&mf&&3.0000&/span&
&span class=&mf&&4.0000&/span&
&span class=&mf&&5.0000&/span&
&span class=&mf&&6.0000&/span&
&span class=&mf&&7.0000&/span&
&span class=&mf&&4.2000&/span&
&span class=&mf&&5.3000&/span&
&span class=&mf&&6.4000&/span&
&span class=&mf&&7.5000&/span&
&span class=&mf&&8.6000&/span&
&span class=&mf&&5.0000&/span&
&span class=&mf&&6.0000&/span&
&span class=&mf&&7.0000&/span&
&span class=&mf&&8.0000&/span&
&span class=&mf&&9.0000&/span&
&span class=&c&&% 操作二：交换两个行&/span&
&span class=&n&&M&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span& &span class=&mi&&2&/span& &span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span&&span class=&p&&;&/span&&span class=&mi&&2&/span& &span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span&&span class=&p&&;&/span&&span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span&&span class=&p&&;&/span&&span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span& &span class=&mi&&8&/span&&span class=&p&&;&/span&&span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span& &span class=&mi&&8&/span& &span class=&mi&&9&/span&&span class=&p&&]&/span&
&span class=&n&&E2&/span& &span class=&p&&=&/span& &span class=&nb&&eye&/span&&span class=&p&&(&/span& &span class=&mi&&5&/span& &span class=&p&&)&/span&
&span class=&c&&% 生成 5x5 单位矩阵&/span&
&span class=&n&&E2&/span&&span class=&p&&([&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&5&/span&&span class=&p&&],:)&/span& &span class=&p&&=&/span& &span class=&n&&E2&/span&&span class=&p&&([&/span&&span class=&mi&&5&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&],:)&/span& &span class=&c&&% 交换行 2 和 行5，生成Type ii 初等矩阵&/span&
&span class=&c&&% Type ii 初等矩阵是通过上述操作生成的&/span&
&span class=&n&&E2&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&n&&E2&/span& &span class=&o&&*&/span& &span class=&n&&M&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&span class=&mi&&4&/span&
&span class=&mi&&5&/span&
&span class=&mi&&5&/span&
&span class=&mi&&6&/span&
&span class=&mi&&7&/span&
&span class=&mi&&8&/span&
&span class=&mi&&9&/span&
&span class=&mi&&3&/span&
&span class=&mi&&4&/span&
&span class=&mi&&5&/span&
&span class=&mi&&6&/span&
&span class=&mi&&7&/span&
&span class=&mi&&4&/span&
&span class=&mi&&5&/span&
&span class=&mi&&6&/span&
&span class=&mi&&7&/span&
&span class=&mi&&8&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&span class=&mi&&4&/span&
&span class=&mi&&5&/span&
&span class=&mi&&6&/span&
&span class=&c&&% 操作3：为一整行乘上非零数量&/span&
&span class=&n&&M&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span& &span class=&mi&&2&/span& &span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span&&span class=&p&&;&/span&&span class=&mi&&2&/span& &span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span&&span class=&p&&;&/span&&span class=&mi&&3&/span& &span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span&&span class=&p&&;&/span&&span class=&mi&&4&/span& &span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span& &span class=&mi&&8&/span&&span class=&p&&;&/span&&span class=&mi&&5&/span& &span class=&mi&&6&/span& &span class=&mi&&7&/span& &span class=&mi&&8&/span& &span class=&mi&&9&/span&&span class=&p&&]&/span&
&span class=&n&&E3&/span& &span class=&p&&=&/span& &span class=&nb&&eye&/span&&span class=&p&&(&/span& &span class=&mi&&5&/span& &span class=&p&&)&/span&
&span class=&c&&% 生成 5x5 单位矩阵&/span&
&span class=&n&&E3&/span&&span class=&p&&(&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span& &span class=&p&&=&/span& &span class=&mf&&1.271543&/span& &span class=&c&&% 生成Type iii 初等矩阵&/span&
&span class=&c&&% Type iii 初等矩阵是将对角线上的一个项换位非零非一的数值&/span&
&span class=&n&&E3&/span& &span class=&p&&=&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.2715&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&n&&E3&/span& &span class=&o&&*&/span& &span class=&n&&M&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mf&&1.0000&/span&
&span class=&mf&&2.0000&/span&
&span class=&mf&&3.0000&/span&
&span class=&mf&&4.0000&/span&
&span class=&mf&&5.0000&/span&
&span class=&mf&&2.5431&/span&
&span class=&mf&&3.8146&/span&
&span class=&mf&&5.0862&/span&
&span class=&mf&&6.3577&/span&
&span class=&mf&&7.6293&/span&
&span class=&mf&&3.0000&/span&
&span class=&mf&&4.0000&/span&
&span class=&mf&&5.0000&/span&
&span class=&mf&&6.0000&/span&
&span class=&mf&&7.0000&/span&
&span class=&mf&&4.0000&/span&
&span class=&mf&&5.0000&/span&
&span class=&mf&&6.0000&/span&
&span class=&mf&&7.0000&/span&
&span class=&mf&&8.0000&/span&
&span class=&mf&&5.0000&/span&
&span class=&mf&&6.0000&/span&
&span class=&mf&&7.0000&/span&
&span class=&mf&&8.0000&/span&
&span class=&mf&&9.0000&/span&
&span class=&c&&% 初等矩阵可逆，其逆亦初等&/span&
&span class=&n&&inv&/span&&span class=&p&&(&/span&&span class=&n&&E1&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&o&&-&/span&&span class=&mf&&0.1000&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&n&&inv&/span&&span class=&p&&(&/span&&span class=&n&&E2&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&n&&inv&/span&&span class=&p&&(&/span&&span class=&n&&E3&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&0.7864&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mf&&1.0000&/span&
&span class=&c&&% row echelon matrix 阶梯形矩阵 非唯一；matlab 有reduced ref 函数，返回唯一值&/span&
&span class=&n&&rref&/span&&span class=&p&&(&/span&&span class=&n&&M&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&0&/span&
&span class=&o&&-&/span&&span class=&mi&&1&/span&
&span class=&o&&-&/span&&span class=&mi&&2&/span&
&span class=&o&&-&/span&&span class=&mi&&3&/span&
&span class=&mi&&0&/span&
&span class=&mi&&1&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&span class=&mi&&4&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&span class=&mi&&0&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&八、转置矩阵&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&Y&/span&&span class=&o&&'&/span& &span class=&c&&% transpose，将行列互换&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&1&/span&
&span class=&mi&&4&/span&
&span class=&mi&&7&/span&
&span class=&mi&&2&/span&
&span class=&mi&&5&/span&
&span class=&mi&&8&/span&
&span class=&mi&&3&/span&
&span class=&mi&&6&/span&
&span class=&mi&&9&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&九、行列式（II）&/b&&/p&&p&&img src=&https://www.zhihu.com/equation?tex=%5Cdet+%5Ba%5D+%3D+a%5C%5C& alt=&\det [a] = a\\& eeimg=&1&&&/p&&p&&img src=&https://www.zhihu.com/equation?tex=%5Cdet+%5Cbegin%7Bbmatrix%7D+a+%26+b+%5C%5C+c+%26+d+%5Cend%7Bbmatrix%7D+%3D+ad+-+bc%5C%5C& alt=&\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc\\& eeimg=&1&&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&A&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&3&/span&&span class=&p&&];&/span&
&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&A&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mi&&3&/span&
&span class=&n&&B&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&;&/span&&span class=&mi&&3&/span&&span class=&p&&,&/span&&span class=&mi&&4&/span&&span class=&p&&];&/span&
&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&B&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&o&&-&/span&&span class=&mi&&2&/span&
&span class=&n&&C&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&3&/span&&span class=&p&&;&/span&&span class=&mi&&4&/span&&span class=&p&&,&/span&&span class=&mi&&5&/span&&span class=&p&&,&/span&&span class=&mi&&6&/span&&span class=&p&&;&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&4&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&];&/span&
&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&C&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mi&&15&/span&
&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&E1&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mi&&1&/span&
&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&E2&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&o&&-&/span&&span class=&mi&&1&/span&
&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&E3&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&mf&&1.2715&/span&
&/code&&/pre&&/div&&p&&br&&/p&&p&&b&十、矩阵置换和余因子矩阵&/b&&/p&&div class=&highlight&&&pre&&code class=&language-matlab&&&span&&/span&&span class=&n&&v&/span& &span class=&p&&=&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&,&/span&&span class=&mi&&3&/span&&span class=&p&&,&/span&&span class=&mi&&4&/span&&span class=&p&&,&/span&&span class=&mi&&5&/span&&span class=&p&&];&/span&
&span class=&nb&&perms&/span&&span class=&p&&(&/span&&span class=&n&&v&/span&&span class=&p&&)&/span& &span class=&c&&% 置换 permutation&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span&
&span class=&mi&&5&/span&
&span class=&mi&&4&/span&
&span class=&mi&&3&/span&
&span class=&mi&&2&/span&
&span class=&mi&&1&/span&
&span class=&mi&&5&/span&
&span class=&mi&&4&/span&
&span class=&mi&&3&/span&
&span class=&mi&&1&/span&
&span class=&mi&&2&/span&
&span class=&mi&&5&/span&
&span class=&mi&&4&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&span class=&mi&&1&/span&
&span class=&mi&&5&/span&
&span class=&mi&&4&/span&
&span class=&mi&&2&/span&
&span class=&mi&&1&/span&
&span class=&mi&&3&/span&
&span class=&mi&&5&/span&
&span class=&mi&&4&/span&
&span class=&mi&&1&/span&
&span class=&mi&&3&/span&
&span class=&mi&&2&/span&
&span class=&mi&&5&/span&
&span class=&mi&&4&/span&
&span class=&mi&&1&/span&
&span class=&mi&&2&/span&
&span class=&mi&&3&/span&
&span class=&mi&&5&/span&
&span class=&mi&&3&/span&
&span class=&mi&&4&/span&
&span class=&mi&&2&/span&
&span class=&mi&&1&/span&
&span class=&mi&&5&/span&
&span class=&mi&&3&/span&
&span class=&mi&&4&/span&
&span class=&mi&&1&/span&
&span class=&mi&&2&/span&
&span class=&mi&&5&/span&
&span class=&mi&&3&/span&
&span class=&mi&&2&/span&
&span class=&mi&&4&/span&
&span class=&mi&&1&/span&
&span class=&mi&&5&/span&
&span class=&mi&&3&/span&
&span class=&mi&&2&/span&
&span class=&mi&&1&/span&
&span class=&mi&&4&/span&
&span class=&n&&cofactor&/span& &span class=&p&&=&/span& &span class=&p&&@(&/span&&span class=&n&&A&/span&&span class=&p&&,&/span& &span class=&n&&r&/span&&span class=&p&&,&/span& &span class=&n&&c&/span&&span class=&p&&)&/span& &span class=&p&&(&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span&&span class=&p&&)&/span&^&span class=&p&&(&/span&&span class=&n&&r&/span&&span class=&o&&+&/span&&span class=&n&&c&/span&&span class=&p&&)&/span&&span class=&o&&*&/span&&span class=&n&&det&/span&&span class=&p&&(&/span&&span class=&n&&A&/span&&span class=&p&&([&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&r&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span& &span class=&n&&r&/span&&span class=&o&&+&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&k&&end&/span&&span class=&p&&],&/span& &span class=&p&&[&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&n&&c&/span&&span class=&o&&-&/span&&span class=&mi&&1&/span& &span class=&n&&c&/span&&span class=&o&&+&/span&&span class=&mi&&1&/span&&span class=&p&&:&/span&&span class=&k&&end&/span&&span class=&p&&]));&/span&
&span class=&n&&cofactor&/span&&span class=&p&&(&/span&&span class=&n&&M&/span&&span class=&p&&,&/span&&span class=&mi&&1&/span&&span class=&p&&,&/span&&span class=&mi&&2&/span&&span class=&p&&)&/span&
&span class=&nb&&ans&/span& &span class=&p&&=&/span& &span class=&o&&-&/span&&span class=&mf&&1.6435e-31&/span&
&/code&&/pre&&/div&&p&&img src=&https://www.zhihu.com/equation?tex=%5Ctext%7Bcof%7D%5C+%28A%29_%7Bi%2Cj%7D%3D%28-1%29%5E%7Bi%2Bj%7D%5Cdet+A_%7Bji%7D& alt=&\text{cof}\ (A)_{i,j}=(-1)^{i+j}\det A_{ji}& eeimg=&1&&&/p&&p&&br&&/p&&p&&b&定理3&/b&&/p&&p&设 &img src=&https://www.zhihu.com/equation?tex=A& alt=&A& eeimg=&1&& 为 &img src=&https://www.zhihu.com/equation?tex=n%5Ctimes+n& alt=&n\times n& eeimg=&1&& 矩阵，使 &img src=&https://www.zhihu.com/equation?tex=C%3D%5Ctext%7Bcof%7D%5C+A& alt=&C=\text{cof}\ A& eeimg=&1&& ， &img src=&https://www.zhihu.com/equation?tex=%5Calpha+%3D+%5Cdet+A& alt=&\alpha = \det A& eeimg=&1&& 。如果 &img src=&https://www.zhihu.com/equation?tex=%5Calpha& alt=&\alpha& eeimg=&1&& 不为零，那么 &img src=&https://www.zhihu.com/equation?tex=A& alt=&A& eeimg=&1&& 是可逆的，且 &img src=&https://www.zhihu.com/equation?tex=CA+%3D+AC+%3D+%5Calpha+I& alt=&CA = AC = \alpha I& eeimg=&1&& 。&/p&&p&&/p&&p&&/p&
语料库语言学基础知识写作计划：矩阵（Matlab, Python, R, ）概率论（, 2）统计学（Matlab, Python, R, Haskell）机器学习（Matlab, Python, R, Haskell） 主要参考资料：Michael Artin 的 Algebra如果没有购买 Matlab，可以下载…
&figure&&img src=&https://pic1.zhimg.com/v2-c6c04fcbc91cf65cb9a360c_b.jpg& data-rawwidth=&640& data-rawheight=&494& class=&origin_image zh-lightbox-thumb& width=&640& data-original=&https://pic1.zhimg.com/v2-c6c04fcbc91cf65cb9a360c_r.jpg&&&/figure&&p&&b&前言：&/b&&/p&&p&这是职业蛙暑期在知乎上的福利文！希望大家看完之后都有所收获！&/p&&p&&br&&/p&&p&&b&{&/b&每到寒暑假，大二大三的同学们就开始忙着找实习了。但是很多人在找实习的时候，其实非常迷茫，不知道自己的专业能去哪里实习。有部分大学生可能不喜欢或不擅长自己的专业，想要跨专业去找实习，又会被“没有专业知识和技能”所限制，更有很多人吐槽自己的专业基本属于“没专业”的状态。&/p&&p&这些在找实习，同时也是毕业后找工作会遇到的“专业”问题，职业蛙愿意在知乎上和大家分享一些过来人的经验与教训，希望能帮大家拨开迷雾，&b&看到更多选择的可能性。}&/b&&/p&&p&&br&&/p&&blockquote&&b&第一篇就来谈一谈“万金油”专业——英语专业吧。&/b&&/blockquote&&p&职业蛙调研采访了96位英语专业的同学（包括英语专业、商务英语专业、英语教育专业、旅游英语专业、英语翻译专业、英语语言文学专业、英美文化研究专业等各个细分方向；也有的同学是曾经学过英语专业，后来转专业或跨专业考研学了其他专业的），选出了四个有代表性的英专同学的实习和就业案例，给大家一些参考。&/p&&p&四个详细举例的英语专业毕业生在校情况和现在工作分别是：&/p&&p&l 英语系学霸+考霸：自由职业的译员，要有一颗“大心脏”&/p&&p&l 成绩中上，目标明确：某985高校下属二本院校英语讲师&/p&&p&l 成绩一般的英语系风云人物：top10国际公关公司&/p&&p&l 成绩一般的小透明：从公务员到外贸到德国百年企业总裁助理&/p&&p&（如果有和你情况相符的，你可以直接下拉到对应的小标题下看详细内容）&/p&&p&&br&&/p&&p&&b&文章较长，如果你没有兴趣看下去的话，可以先看这个结论。&/b&&/p&&p&&b&总的来说，只要不是工科、医科、法律（学法律英语除外）等特别“专业”的工作，英语专业的同学真的都可以去做，尤其是涉外的公司，比如国际广告、公关公司、500强等。&/b&&/p&&p&&br&&/p&&blockquote&&b&英语专业可以找的实习工作有：&/b&&/blockquote&&p&l 翻译（笔译、陪同翻译、口译）&/p&&p&l 老师/助教（培训机构）&/p&&p&l 编辑/记者（新媒体、网站、出版社、杂志报纸）&/p&&p&l 文案/策划（广告、市场、公关、会展）&/p&&p&l 销售/推广（外贸、渠道拓展）&/p&&p&l 人力资源（招聘、薪资、企业文化）&/p&&p&l 客服（所有）&/p&&p&l 部门助理/总监/总裁助理（所有）&/p&&blockquote&&b&英语专业毕业后的就业方向有：&/b&&/blockquote&&p&l 翻译&/p&&p&l 老师/助教&/p&&p&l 编辑/记者&/p&&p&l 文案/策划&/p&&p&l 销售/推广&/p&&p&l 人力资源&/p&&p&l 四大&/p&&p&l 客服&/p&&p&l 助理&/p&&p&l 银行&/p&&p&l 导游&/p&&p&l 主持人&/p&&p&l 公务员&/p&&p&&br&&/p&&blockquote&&b&&1&英语系学霸+考霸：自由职业的译员，要有一颗“大心脏”&/b&&/blockquote&&p&Rachel是国内某外国语大学英语专业科班出身，从大一入学到大四毕业一直都是学霸，成绩没掉出过前三。不仅如此，她的目标非常明确，毕业后要做翻译，所以从大二开始就把国内各种翻译考试都考了个遍，每个都顺利通过，名副其实的“考霸”，手握CATTI二级笔译、二级口译、上海高级口译等各种证书。&/p&&p&但是，实习经验是她的弱项。&b&毕业前，她的翻译（口译）经历只有在一些活动中当志愿者&/b&，比如上海世博会、天津达沃斯等。其余的都是&b&跟着学长学姐做笔译，翻译过一些热门的英语畅销小说和外资公司的产品手册&/b&。&/p&&p&毕业那年，Rachel加入了一家翻译公司，通过公司接的活儿，参与各种会议的翻译，外国客户参观的陪同翻译，从工程机械到地质能源，从金融证券到电子商务。所有的专业知识（英文的）都是从零学起，而且必须在短时间内快速融会贯通，前期学习以及记忆的压力是非常大的。一开始她也是给“老师”打下手、准备资料，自己跟着练习。后来从一些小型的非正式的调研、沙龙、座谈做起，积累经验（包括临场经验、行业专业知识、翻译技能等），在已经有了500场以上的会议经验，积累了一些资源之后，她现在可以独立接活，做了一名自由职业者。&/p&&figure&&img src=&https://pic3.zhimg.com/v2-d18d3b6a94b861b1ce5c_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&554& data-rawheight=&340& class=&origin_image zh-lightbox-thumb& width=&554& data-original=&https://pic3.zhimg.com/v2-d18d3b6a94b861b1ce5c_r.jpg&&&/figure&&p&
图为外交部翻译张璐（现任外交部翻译司西葡语处处长）&/p&&p&&b&【Rachel忠告】&/b&英专的学弟学妹们，如果真的想走译员这条路，从大一开始就要做好准备了，而且不能朝三暮四，翻译训练一旦中断，哪怕只有一个月，水平也会大幅下降。&/p&&p&&b&【职业蛙知乎福利文小建议】&/b&&/p&&p&1）把自己“逼疯”，最好达到“走火入魔”的地步——哪怕是听到地铁上有老外在聊天，也要自动翻译成中文，看到各种行业报告、新闻要自动翻译成英文；&/p&&p&2）积累各行业的知识，尤其是经常有举行国际会议、论坛的行业，比如经济、能源、机械、化工、医学、建筑、互联网等；&/p&&p&3）保持一颗好奇心，新的领域、新的技术、新的网络用语都要了解一点，历史典故、名人轶事、科学理论也都要知道一点，发言嘉宾的知识范围之大永远超出你的想象——一个讲营销的会，嘉宾用了10分钟来介绍爱迪生和特斯拉之间的恩怨情仇，就为了说明爱迪生是个营销大师；&/p&&p&4）还有就是要练就一颗“大心脏”，毕竟在活动中随时都有可能出现演讲嘉宾的发言“货不对板”的情况——医学行业的研讨会，开场嘉宾临时换人讲了40分钟的“科学发展观”，也不是个别情况；&/p&&p&5）坚持锻炼身体，保持充沛的体力，能保持一定时间（至少20分钟）的高度精神集中和快速反应——不信的话，可以试试连续做30分钟新闻的shadowing practice，没有经过训练，就能在30分钟里一次都不走神算我输；&/p&&p&6）准备两套有质感的职业套装，学会化符合自己气质的淡妆；&/p&&p&7）准备好所有需要的证件：护照、港澳通行证、入台证——在大学期间提前办好这些证件，有备无患。曾经有过被临时通知去香港参加个会，然后因为没有港澳通行证而没有去成，失去了一次好机会。&/p&&p&&br&&/p&&blockquote&&b&&2&成绩中上，目标明确：某985高校下属二本院校英语讲师&/b&&/blockquote&&p&Sophia的本科就读于国内排名前三的外国语大学，本科毕业后去英国读了教育学的研究生，回国后先是在某知名英语教育机构当了近一年的雅思老师，后来恰逢某985高校下属的二本院校招聘英语老师，成功获聘。&/p&&p&在大学期间，Sophia的就业目标一直很明确，就是想当老师，只不过纠结于当中小学老师还是大学老师。所以，她在大学的时候就考了教师资格证、TESOL等。在学校期间的成绩也始终保持在专业前十，并获得了优秀论文（原话是：如果以后当了老师，我的论文都是学生们可以查的到的，要以身作则好好写）。&/p&&p&&b&实习经历来说，Sophia在各大培训机构做过老师，从新概念英语到四六级考试，从雅思到商务英语，从一年级小朋友到40岁成功人士，她都教过。培训班除了教课以外，还需要做教案、课程大纲，刷题、分析出题思路、押题等。比较特别的是，她还在某个直播平台撒钱奖励老师入驻的时候，短暂开过直播课，梦想当一回“网红”，也了解一些在线教育的方法。&/b&&/p&&p&这些实习经历对于她后来去面试大学的英语讲师来说都是加分项。再加上她的英国留学经历和本科学校不错，又能和学生打成一片，所以现在在学校里也是干得风生水起。&/p&&figure&&img src=&https://pic2.zhimg.com/v2-d0ffeed61cf43a41710efbd253b94f1f_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&440& data-rawheight=&457& class=&origin_image zh-lightbox-thumb& width=&440& data-original=&https://pic2.zhimg.com/v2-d0ffeed61cf43a41710efbd253b94f1f_r.jpg&&&/figure&&p&
图为川师文理学院“最美英语老师”&/p&&p&&b&【Sophia忠告】&/b&英专的同学，尤其是考研的，大多数都想过当老师，或者把当英语老师作为自己的“底线”——大不了找不到工作，我就去培训机构当英语老师么，这种话说的人特别多。但其实现在培训机构也不像以前这么好进了。对学历、留学经历、教学经验、教学方式，偶尔还有颜值，都有要求。听说，现在很多进培训机构当英语老师的英专同学都是从教学助理做起，先积累经验才有机会去讲课。培训机构的老师的话，大多数都是给课时费的，如果课时少的话，相对而言收入也会少，要熬到资深，获得更多讲课机会才有钱途。&/p&&p&&b&【职业蛙知乎福利文小建议】&/b&&/p&&p&1）多看一些网上的公开课，看看别人是怎么把课上的有趣的；&/p&&p&2）教学大纲的变化是不多的，但是教学内容，尤其是举例可以与时俱进，比如最新的演讲、电影、脱口秀等等都是很好的素材；&/p&&p&3）不要停止学习，多开拓眼界，你的学生可能每个寒暑假都在英国美国游学，她们懂的真的比你多；&/p&&p&4）大学里学的是思维方式和学习能力，这是终身有用的。&/p&&p&&br&&/p&&blockquote&&b&&3&成绩一般的英语系风云人物：top10国际公关公司&/b&&/blockquote&&p&Yolanda可能是天生的公关人，在学校里虽然成绩一般，但却是学院学校的名人。她积极参加各种社团活动：学生会、演讲社、话剧社、电视台，而且承担不同的职责，有外联、有策划、有记者、有主持、有演员。学校里的每一个大型活动都有她的参与，上至政府领导参观学校的陪同介绍，下到每个寝室敲门发传单为自己的话剧做宣传。&/p&&p&除了学校的活动以外，&b&她的实习经历也非常丰富：4A广告公司AAE、会展公司小文案、公关公司媒介实习生、网站和新媒体小编。她知道自己的优势是擅长与人沟通，擅长演讲，反应快、有一点点英语优势，但是她不确定自己最想要做的工作是什么。所以，她在大学期间尝试了各种职能的工作，通过在工作中观察、体验，找准自己的定位。&/b&&/p&&p&现在，Yolanda就职于全球排名前十的公关公司，服务的客户也都是跨国企业，做过的公关活动也常常在朋友圈刷屏。虽然，每天忙到爆炸，欣赏凌晨四点的风景也是家常便饭，但她很喜欢自己的工作，很有满足感和成就感。&/p&&figure&&img src=&https://pic3.zhimg.com/v2-1540cec727cef8acc7be9_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&554& data-rawheight=&233& class=&origin_image zh-lightbox-thumb& width=&554& data-original=&https://pic3.zhimg.com/v2-1540cec727cef8acc7be9_r.jpg&&&/figure&&p&
图为美剧Suits中Rachel的造型，值得初入职场的公关小伙伴借鉴&/p&&p&&b&【Yolanda忠告】&/b&公关公司真的有很多英语专业的女生，语言能力在新人阶段真的是有一点优势的，毕竟服务的客户都是跨国公司，英语就是工作语言。同样都是什么都不懂的新人，英语好一点的就会更自信一点，而且上手会更快一点。但是，最重要的还是专业能力，所以想要进公关公司的同学，在学校的时候就可以刻意练习培养一些公关工作必须的技能和习惯了。&/p&&p&&b&【职业蛙知乎福利文小建议】&/b&&/p&&p&1）沟通能力真的很重要，尤其是“简单扼要地说人话”的能力。公关需要沟通的一般都是大忙人，自己的老板、客户的老板、媒体、政府官员，能用一分钟讲清楚的事情就别用一分半，必须要有“用最短的时间把事情讲清楚”的能力，强烈建议大家去学一下金字塔原理，从结论出发，逻辑清楚地把事情讲明白。&/p&&p&2）写作能力。很多做公关的以前是做记者的，无论是老板的发言稿、活动的新闻稿、公司的声明、项目的展示汇报等等，都需要极强的写作能力。如果英语专业的同学真的想走公关这条路，在学校的时候就可以练习起来了。比如现在很多发布会都是网络直播的，看完直播后，你就可以以品牌公关的角度写新闻稿了，再拿你写的和真正公司通发的新闻稿，以及记者写的稿件做对比，看看差在哪里。多写多练，自然就能掌握一定的技巧和套路。中英文的写作能力都需要练习起来。&/p&&p&3）多任务同时处理的能力，在学校里的一边刷题一边看美剧不叫多任务处理，只能叫分心。工作中，可能同时跟进四五个项目，每个项目的阶段不同、需求不同、难点不同、配合的部门和同事不同，要把握好每个项目的节奏和细节，整个人要像电脑一样高速运转，不能顾了这头忘了那头；&/p&&p&4）什么都要会一点：做公关真的要把自己当全才来培养，翻译临时没来要自己上，设计师忙不过来要自己作图修图，摄影师迟到了要自己拍照，主持人老板不满意要拿着话筒就上台，甚至现场座位摆得间距太小，没人帮忙要自己来一个个摆，所有光鲜亮丽的背后都是看不到的付出；&/p&&p&5）形象管理能力，这个是我觉得英语专业女生有优势的地方，一般来说英专的学生都会化个淡妆、做个发型，穿衣打扮也比较时尚得体。不是说每个人都要像电影电视里那样从头到脚穿名牌，但是毕竟是代表公司的形象，准备几套得体的、品质好的职业装还是很有必要的，这个投资不要省，可以参考《我的前半生》里唐晶的风格（虽然有点过于稳重，但不会出错），也可以参考美剧《Suits》里Rachel的风格（她在剧里是法律助理，比较初级的职位，但她出身名门，所以穿衣风格很有品位。小八卦：演Rachel的演员本人正在和哈里王子谈恋爱~~八卦能力也是公关必备，哈哈哈！）。&/p&&p&&br&&/p&&blockquote&&b&&4&成绩一般的小透明：从公务员到外贸到德国百年企业总裁助理&/b&&/blockquote&&p&Laura是某985理工科大学的英语系“小透明”，当然在学校里英语系本身也不太受待见，参与各种学校活动的机会都比较少。Laura本人也是一个内向求稳的人，所以在大学里成绩一般般，表现一般般，人缘一般般，除了点名的时候，基本上没有人会记得她。&/p&&p&&b&实习经历的话，Laura也没有特别大的志向一定要找500强或者外资企业，简历是海投的，有什么公司找她去面试她就去，有什么公司录用她去实习她就接受。所以，她的实习经历虽然多，但是看不出她的职业规划方向，比如她做过广告公司的文案，做过互联网公司的运营，做过医疗公司的部门助理。大四那年，她也跟风考了公务员、考了研究生，当然都是没考上。&/b&&/p&&p&毕业的时候，她在家人的安排下，回到山东，在自己家的小地方的事业单位过起了“养老的日子”。但是，半年后这种“好日子”就结束了，Laura的爸爸生了一场大病，不仅花光了家里所有的积蓄，还欠了一大笔外债。Laura为了尽快还清欠款，就从每个月收入几千块的事业单位辞职，去了一家外贸公司工作，收入比之前翻一番，当然辛苦程度翻了好几番。&/p&&p&大学期间选修过商务英语的Laura，去了外贸公司工作，才知道学校里学的那叫一个脱节啊，基本上完全用不到。怎么写开发信，怎么写催款信，怎么翻译各种证明资质，怎么和客户打电话，这些基本上都要在工作中现学，所以刚刚开始做外贸的时候特别忙，还要处理“两个时差”工作，一个月不开单，舱位爆满订不到，海关扣货发不了等各种压力和突发状况。&/p&&p&好在Laura熬过了这段最艰难的时候。在外贸公司工作四年，把欠款全部还清，还积累了不少优质客户，其中一家德国公司就特别欣赏她做事认真负责，踏实细致的工作态度，所以就挖了墙角。Laura跳槽到这家德国家族企业，刚开始做的是部门助理的工作，要快速把部门各种杂七杂八的工作都搞清楚，和部门里各种人打交道，协调平衡部门之间的合作。做得好的话，很容易成为部门各种信息的“中转站”，知道的信息可能比老板还多；做得不好的话，可能就要各种做“背锅侠”了。&/p&&p&今年Laura又升职了，做到了总裁助理，老板是德国人。她大学时候学的二外德语终于被证明是“完全无用的”，不过好歹学过所以捡起来还是比从头学要快一点，大多数时候还是用英语交流。总裁助理和部门助理不同，打交道的人从经理、主管、助理变成了总监，参与的会议也从执行方案变成了战略决策，对于这份新的工作，Laura坦言还在学习中。&/p&&figure&&img src=&https://pic3.zhimg.com/v2-5cd7bd7a84c0c9f7f9a63db_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&554& data-rawheight=&360& class=&origin_image zh-lightbox-thumb& width=&554& data-original=&https://pic3.zhimg.com/v2-5cd7bd7a84c0c9f7f9a63db_r.jpg&&&/figure&&p&
图为电影《穿普拉达的女王》主编大大vs小助理&/p&&p&&b&【Laura忠告】&/b&在学校里不要把求稳当成自己的目标，即使你打算一毕业就当公务员甚至当全职家庭主妇，你也要树立一个职业目标，毕竟很多意外是你不可预测的。在学校里学习、实习和试错的成本是最低的。&/p&&p&&b&【职业蛙知乎福利文小建议】&/b&&/p&&p&1）找工作求稳的心态往往是最不稳的；&/p&&p&2）做部门助理对于刚毕业的大学生是一个不错的选择，相对来说要求也低一点，助理的工作其实是一门与人打交道的艺术，需要时间和经验的积累；&/p&&p&3）部门助理做得好的话，会有人脉积累，会比一般人有更多转岗甚至跳槽的机会，也很有可能跟着老板一起平步青云；&/p&&p&4）做外贸很辛苦，压力很大，要和“老司机”搞好关系，她们的很多经验教训可以帮你节省一两年的摸索，很多专业英语是字典里和谷歌里查不到的，只有公司里的“活字典”知道；&/p&&p&5）你英语再好，也很难完全听懂印度人、印尼人、马来人、肯尼亚人的英语，虽然她们都自称英语是母语，电话打完，一定要用邮件再确认一次电话沟通的内容；&/p&&p&6）学校里的二外要好好学，毕业之后也要找机会多使用，说不定什么时候就派的上用场；&/p&&p&7）工作之余多陪父母，定期带父母检查身体，自己也要多锻炼，健康是1，其他都是1后面的0。&/p&&p&&br&&/p&&p&&b&小结：&/b&&/p&&p&职业蛙为了这篇知乎暑期福利文，采访了很多英语专业毕业的同学，工作经验从1年到10年不等，大多数是已经工作超过5年的资深白骨精了。她们还有很多的关于实习、关于工作选择的故事。我们挑选了几个有代表性的故事和恳切的建议，分享给大家，希望对大家有帮助。&/p&&p&另外，职业蛙根据所有采访过的英专同学目前和曾经从事过的职业以及实习过的经历，进行了梳理，一共梳理了8种适合英语专业同学实习的工作和13种适合英语专业同学毕业后就业的工作，可以拉到上面的结论部分查看哦。&/p&&p&&b&这是一篇调研了三个月，花了一周时间来写的纯干货分享知乎文，职业蛙的小蛙们真的是呕心沥血，如果大家觉得有帮助的话，还请大家多多点赞、收藏！&/b&&/p&&p&&br&&/p&&p&职业蛙在知乎上还回答了以下有关内容：&br&&a href=&https://www.zhihu.com/question//answer/& class=&internal&&什么样的应届生值得招？ - 知乎&/a& &a href=&https://www.zhihu.com/question//answer/& class=&internal&&面试前你都做了什么准备？ - 知乎&/a& &a href=&https://www.zhihu.com/question//answer/& class=&internal&&应届毕业生第一份工作的选择有多重要？应该考虑清楚些什么？ - 知乎&/a&&/p&&p&想要获取更多求职咨询请关注职业蛙微信公众号：careerfrog &br&也欢迎各位知乎小主移步职业蛙主页了解更多干货！&/p&&figure&&img src=&https://pic2.zhimg.com/v2-2f8a961290bbadfa83c6a8_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&250& data-rawheight=&250& class=&content_image& width=&250&&&/figure&&p&&/p&&p&&/p&
前言：这是职业蛙暑期在知乎上的福利文！希望大家看完之后都有所收获！ {每到寒暑假，大二大三的同学们就开始忙着找实习了。但是很多人在找实习的时候，其实非常迷茫，不知道自己的专业能去哪里实习。有部分大学生可能不喜欢或不擅长自己的专业，想要跨专业…
&figure&&img src=&https://pic4.zhimg.com/v2-6afdbb932cd_b.jpg& data-rawwidth=&1000& data-rawheight=&431& class=&origin_image zh-lightbox-thumb& width=&1000& data-original=&https://pic4.zhimg.com/v2-6afdbb932cd_r.jpg&&&/figure&&p&备考GRE和托福考试的同学可能会听说过一本叫“韦氏小绿”的书，它的全称是 Merriam-Webster's Vocabulary Builder，它是由美国知名词典商Merriam-Webster 推出的一本针对应试以及词汇提升的词汇书，目前出到第二版。&/p&&p&&br&&/p&&p&相比市面上其他词汇书，“韦氏小绿书”有以下几个特点：&/p&&p&&br&&/p&&p&1. 采用词根的形式将各个词串联在一起，每个单元8个词根，每个词根4个单词，再加上由神话以及历史故事演化来的8个单词，一共40个单词。&/p&&p&2. 书中对单词并不只是简单给出释义，而是带有详细例句以及非常精辟的解释。&/p&&p&3. 书中的Quizzes（检测题）设计得很巧妙。&/p&&p&&br&&/p&&p&在很多人的印象里面，韦氏小绿书似乎就是一本用来记词根词缀的书，但这本书真正的价值并不在于此。韦氏小绿书虽然提供了250个词根，但这些词根在实用价值上并不大。这本书最大的好处在于提供了一个通过语境记单词的方法。&/p&&figure&&img src=&https://pic2.zhimg.com/v2-f9eeed281be7dbddb22df_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1080& data-rawheight=&1920& class=&origin_image zh-lightbox-thumb& width=&1080& data-original=&https://pic2.zhimg.com/v2-f9eeed281be7dbddb22df_r.jpg&&&/figure&&p&&br&&/p&&p&在图书的序言中，作者提到了通过语境记单词的好处：人脑并不擅长记忆分散的没有逻辑联系的事物，只有将单词放在一个个具体的语境中，我们才能理解以及使用它。&/p&&p&&br&&/p&&blockquote&The fact is that we tend to remember words easily and naturally when they appear in some meaningful context, when they have been shown to be useful and therefore worth remembering, and when they have been properly explained to us.&/blockquote&&p&&br&&/p&&p&秉承这一思想，韦氏小绿书对于单词的诠释分为三个部分：英文释义，例句以及作者对于单词的个性化说明。举个例子，对于单词 conducive，作者是这样解读的：&/p&&figure&&img src=&https://pic2.zhimg.com/v2-a1aa8dd8ecfb_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1280& data-rawheight=&881& class=&origin_image zh-lightbox-thumb& width=&1280& data-original=&https://pic2.zhimg.com/v2-a1aa8dd8ecfb_r.jpg&&&/figure&&p&通过例句以及作者的说明，我们可以很清楚地理解conducive的用法：&/p&&p&&br&&/p&&p&1. conducive 一般指“（某件事）有助于某事，有益于某事”，表示积极含义；&/p&&p&2. conducive 后面的介词一般是to，接名词，常见搭配是 be conducive to study/thinking/conversation/sleep 等。&/p&&p&&br&&/p&&p&对于名词fractal（分形），作者则解释了它的来源和应用：&/p&&figure&&img src=&https://pic3.zhimg.com/v2-9dd711a777c9ad67facd4cab_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1280& data-rawheight=&729& class=&origin_image zh-lightbox-thumb& width=&1280& data-original=&https://pic3.zhimg.com/v2-9dd711a777c9ad67facd4cab_r.jpg&&&/figure&&p&&br&&/p&&p&对于来自神话以及历史故事的词汇，书中还会解释单词背后的故事。举个例子，在读外刊或原著的时候我们可能会经常见到Sisyphean这个词，书中给出了这样的解释：&/p&&p&&br&&/p&&figure&&img src=&https://pic3.zhimg.com/v2-72f24993fca83bcd59b269c799c0c6ab_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&3024& data-rawheight=&1255& class=&origin_image zh-lightbox-thumb& width=&3024& data-original=&https://pic3.zhimg.com/v2-72f24993fca83bcd59b269c799c0c6ab_r.jpg&&&/figure&&figure&&img src=&https://pic3.zhimg.com/v2-cd7ff612df4ad76a4f73_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&3024& data-rawheight=&836& class=&origin_image zh-lightbox-thumb& width=&3024& data-original=&https://pic3.zhimg.com/v2-cd7ff612df4ad76a4f73_r.jpg&&&/figure&&p&Sisyphean源自希腊神话：西西弗斯(Sisyphus)因前生罪恶受惩罚，要无休止地将一块巨石滚到山顶，而巨石总是一再滚落。现在Sisyphean一般引申为“（任务）无法完成的，永无休止的”，常见的搭配是 a Sisyphean task.&/p&&p&&br&&/p&&p&除了别具一格的单词解读之外，韦氏小绿人的另一个亮点在于测验题的设计。书中安排了大量的巩固测试题，而且这些测试并不只是简单考单词的含义，而是考察单词在语境中的具体使用以及对形近词的辨析。比如：&/p&&figure&&img src=&https://pic1.zhimg.com/v2-fc8bebdc8a1b381bc5da604b_b.jpg& data-caption=&& data-size=&normal& data-rawwidth=&1280& data-rawheight=&1058& class=&origin_image zh-lightbox-thumb& width=&1280& data-original=&https://pic1.zhimg.com/v2-fc8bebdc8a1b381bc5da604b_r.jpg&&&/figure&&p&ambiguous, ambient, ambivalent, ambit 这几个词看起来有点像，但它们在意思上存在着不少区别，比如 ambiguous 表示协议和言辞等含混不清，而ambivalent则侧重于表示人的心情矛盾，模棱两可。如果不明白这几个区别，上面的练习题是无法完成的。&/p&&p&&br&&/p&&p&韦氏小绿书一共有30个单元，全书包含的词汇在3500词左右。客观上来说这本书的词汇难度并不低，大概在托福和GRE的水平之间。书中绝大部分单词的实用价值都很高，诸如xenophobia, oligarch, per se, plutocracy, loquacious 这类单词都是在严肃阅读和写作中经常会用到的词汇。如果将它们词全部掌握，相信你的学术词汇功底会大大增强。当然，书中也有一些单词较为生僻，部分单词比如 catacomb, atrium, arachnophobia在日常生活中的使用频率非常低，可记可不记。&/p&&p&&br&&/p&&p&小绿书的各个单元彼此独立，因此在阅读时不必按照从头到尾的顺序，可以选择自己感兴趣的单元进行识记。为了保证最好的使用效果，推荐各位在学习单词的释义和例句之后进行造句练习。&/p&&p&&br&&/p&&p&举个例子，对于上面提到的conducive，我们可以仿写两个句子：&/p&&p&&br&&/p&&p&1. Governments should try to provide rural students with an environment &b&conducive to&/b& learning.&/p&&p&&br&&/p&&p&2. A bedroom facing onto a busy street is not &b&conducive to&/b& a good night's sleep.&/p&&p&&br&&/p&&p&这样可以进一步加深对单词的印象。一般来说，经过完成测验，反复识记和造句练习，一个单词会有更大的几率变成我们的主动词汇。&/p&&p&&br&&/p&&p&推荐此书给以下读者：&/p&&p&&br&&/p&&p&1. 具有大学六级左右基础，词汇量在7000左右（词汇量太低书中有些解释可能会看不懂），习惯阅读英语原版读物，且对提升词汇功底感兴趣的同学；&/p&&p&&br&&/p&&p&2. 准备SAT，托福和GRE等出国考试的同学。&/p&&p&&br&&/p&&p&&b&扩展阅读：&/b&&/p&&p&&br&&/p&&p&&a href=&https://link.zhihu.com/?target=https%3A//www.douban.com/group/topic//& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&为什么你需要多读英语原版书？&/a&&/p&&p&&a href=&https://link.zhihu.com/?target=https%3A//www.douban.com/group/topic//& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&怎样测试自己的词汇量？&/a&&/p&
备考GRE和托福考试的同学可能会听说过一本叫“韦氏小绿”的书，它的全称是 Merriam-Webster's Vocabulary Builder，它是由美国知名词典商Merriam-Webster 推出的一本针对应试以及词汇提升的词汇书，目前出到第二版。 相比市面上其他词汇书，“韦氏小绿书”…
太多了，随便举几个例子吧。补充下：以下都是美国人跟我讲的，而且很多都是美式用法。&br&&br&1 电脑游戏不叫computer game,地道的说法是video game（补充：One is on the computer and the other is on the tv, but in spoken english most call both video games.刚问完美国人）&br&&br&2 酒可以用booze来表示，不过是指喝的酒比较多的情况下。&br&比如：I drank way too much booze last night.&br&&br&3
钱不一定用dollar可以用buck来表示&br&Do u have a few bucks u can loan me.&br&&br&4 形容好听的音乐用groovy music&br&&br&5晒太阳用catch some rays比bask更地道&br&&br&6 墨镜可以用shades&br&Those are really cool shades u r wearing!&br&&br&7 打篮球可以不用play basketball,用shoot hoops会更好&br&&br&8擅长可以用pro&br&She is really a pro at basketball.&br&&br&9喧闹可以不用noise 用racket&br&Small kids can really make a lot of racket.&br&&br&10 形容抽烟，吸毒很嗨的状态用stoned&br&&br&11对方说话没听清很少用pardon 而要说Sorry，What did u say?&br&&br&12 被什么打鸡血了可以说pumped(up)&br&I'm really pumped for the holiday break.&br&&br&13 糟糕可以用suck替代terrible&br&The whole trip sucked.&br&&br&14 失败可以用bust替代failure&br&It was a total bust.&br&&br&15 solid 这个词很活 &br&形容连续的几天可以用solid days &br&It's been raining for seven solid days.&br&&br&16 朋友可以用pal来代替friend（评论区有人指正：pal 有点发小、哥们儿的意思，一般是同性之间称呼。）&br&&br&17 catch u later可以替代see u later&br&&br&先写这些，想到继续补充。
太多了，随便举几个例子吧。补充下：以下都是美国人跟我讲的，而且很多都是美式用法。 1 电脑游戏不叫computer game,地道的说法是video game（补充：One is on the computer and the other is on the tv, but in spoken english most call both video games…
&p&&b&先看视频：&/b&&/p&&br&&p&&a class=&video-box& href=&//link.zhihu.com/?target=https%3A//v.qq.com/x/page/e03477lqj43.html& target=&_blank& data-video-id=&& data-video-playable=&& data-name=&TraveliD - 看片识地儿 - 腾讯视频& data-poster=&http://shp.qpic.cn/qqvideo_ori/0/e03477lqj43_228_128/0& data-lens-id=&&&
&img class=&thumbnail& src=&http://shp.qpic.cn/qqvideo_ori/0/e03477lqj43_228_128/0&&&span class=&content&&
&span class=&title&&TraveliD - 看片识地儿 - 腾讯视频&span class=&z-ico-extern-gray&&&/span&&span class=&z-ico-extern-blue&&&/span&&/span&
&span class=&url&&&span class=&z-ico-video&&&/span&https://v.qq.com/x/page/e03477lqj43.html&/span&
&/a& （字幕由&a href=&//www.zhihu.com/people/192fe28683fcb25f7aa5e& data-hash=&192fe28683fcb25f7aa5e& class=&member_mention& data-title=&@TianciHuang& data-editable=&true& data-hovercard=&p$b$192fe28683fcb25f7aa5e&&@TianciHuang&/a&制作）&/p&&br&&p&&b&视频背景：&/b&&/p&&br&&p&第一期的视频来自于 Youtube 知名博主 &a href=&//link.zhihu.com/?target=https%3A//www.youtube.com/channel/UCtinbF-Q-fVthA0qrFQTgXQ& class=& wrap external& target=&_blank& rel=&nofollow noreferrer&&CaseyNeistat&/a&，他这这个广告片是受 Nike 的邀请，帮助拍摄一部关于nike手环的广告语 Make it count 的视频。这小子很聪明，拿了他们的预算，花了10天时间去环游世界，拍出来短片在 Youtube 上快速获得了2000多万的播放量。当然，对于国人来说，Casey这小子更被国人所知晓和疯传的是这个：&a hr}