Gaussian Elimination

The new coefficients are $$\begin{equation} b_{1k} = a_{1k}^{(1)} \quad k=1,\dots,n, \tag{14} \end{equation}$$ where each $a_{1k}^{(1)}$ is equal to the original $a_{1k}$ element. The other coefficients are $$\begin{equation} a_{jk}^{(2)} = a_{jk}^{(1)}-\frac{a_{j1}^{(1)}a_{1k}^{(1)}}{a_{11}^{(1)}} \quad j,k=2,\dots,n, \tag{15} \end{equation}$$ with a new right-hand side given by $$\begin{equation} y_{1}=w_1^{(1)}, \quad w_j^{(2)} =w_j^{(1)}-\frac{a_{j1}^{(1)}w_1^{(1)}}{a_{11}^{(1)}} \quad j=2,\dots,n. \tag{16} \end{equation}$$ We have also set $w_1^{(1)}=w_1$, the original vector element. We see that the system of unknowns $x_1,\dots,x_n$ is transformed into an $(n-1)\times (n-1)$ problem.

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