Linear system/Matrix/Remark

If we multiply an ${}m\times n$-matrix ${}A=(a_{ij})_{ij}$ with a column vector ${}x={\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}$, then we get

${}Ax={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}\end{pmatrix}}\,.$

Hence, an inhomogeneous system of linear equations with disturbance vector ${}{\begin{pmatrix}c_{1}\\c_{2}\\\vdots \\c_{m}\end{pmatrix}}$ can be written briefly as

${}Ax=c\,.$

Then, the manipulations on the equations that do not change the solution set, can be replaced by corresponding manipulations on the rows of the matrix. It is not necessary to write down the variables.