一、线性方程组概念
1、一般我们所说的线性方程组,一般有未知数(一次)、系数、等号等组成,如下所示:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/a9d3fd1f4134970a756d44309acad1c8a7865d96?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
2、线性方程组可以转化成矩阵形式,如下所示:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/ca1349540923dd546443f377de09b3de9c8248f8?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
3、将等式右端,加入矩阵,形成增广矩阵能有效的求出线性方程组的解,如下:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/d4628535e5dde711897c17e9a8efce1b9d166106?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
二、方程组的通解
1、方程组还可以写成如下所示的向量形式:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/0ff41bd5ad6eddc4368de0e436dbb6fd52663342?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
2、方程组通解的概念:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/242dd42a2834349be7fe3a9ac6ea15ce36d3be31?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
3、求方程组通解的基本方法,一般有换位变换,数乘变换,倍加变换等,如下:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/c8ea15ce36d3d539500f5fdc3587e950352ab0fb?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
三、行阶梯方程
1、利用初等行变换求解以下方程组:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/6609c93d70cf3bc7d2f99cf3de00baa1cd112a37?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
2、化简为行阶梯方程组:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/0824ab18972bd4075b2282e274899e510eb309c3?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
3、行阶梯方程组概念,如下图所示。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/1f178a82b9014a9067f817a2a6773912b21beeef?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
四、经典例题——求通解
1、求解下题方程组的通解:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/6d81800a19d8bc3e8bf357d58d8ba61ea8d34531?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
2、转换成,行阶梯方程组,并定义自由未知数,因此,可以得出该题通解,如下:
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/6c224f4a20a44623ceb4f57d9722720e0cf3d7b7?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)