$$ \left\{ \begin{array}{lllll} a_{11}x_1 &+a_{12}x_2&+ \ldots&+a_{1n}x_n &=b_1,\\ a_{21}x_1 &+a_{22}x_2&+ \ldots&+a_{2n}x_n &=b_2,\\ \dots & & & & \dots \\ a_{m1}x_1 &+a_{m2}x_2&+ \ldots&+a_{mn}x_n &=b_m. \end{array} \right. \quad {\color{Red}{\iff}} \quad A X = \mathcal B \, , \ \operatorname{rank} A = {\color{Brown}{\mathfrak r }} $$

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