#mathematic #linear-algebra # [[Epistemic status]] #shower-thought #to-digest # Changelog ```dataview TABLE WITHOUT ID file.mtime AS "Last Modified" FROM [[#]] SORT file.mtime DESC LIMIT 3 ``` # Related # TODO > [!TODO] TODO # Gaussian elimination The Gaussian elimination uses the following formula: $ x_j = x_j - \frac{\alpha_{ij}}{\alpha_{ii}}x_i $ $ \alpha_{ij} = \alpha_{ij} - \frac{\alpha_{ii}}{\alpha_{ij}}\alpha_{ij} $ So to change the $i$-th row with the $j$-th row we need to do the following: $ \alpha_{ij} = \alpha_{ij} - \frac{\alpha_{ \ell}}{\alpha_{ \ell}} \alpha_{i \ell} - \frac{\alpha_{ \ell}}{\alpha_{ \ell}} \alpha_{\ell j}. $ It is not difficult to see that this is equivalent to $ \alpha_{ij} = \alpha_{ij} - \frac{\alpha_{ \ell}}{\alpha_{ \ell}} \alpha_{\ell j} $ and in particular $\alpha_{ij} = 0$ if $i \neq j$.