Matrix/R/Mapping/Example

Let an ${\displaystyle {}m\times n}$-matrix

${\displaystyle {\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}}$

be given, where the entries ${\displaystyle {}a_{ij}}$ are real numbers. Such a matrix defines a mapping

${\displaystyle \varphi \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m},}$

by sending an ${\displaystyle {}n}$-tuple ${\displaystyle {}x={\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}\in \mathbb {R} ^{n}}$ to the ${\displaystyle {}m}$-tuple

${\displaystyle {}\varphi (x)={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{j=1}^{n}a_{1j}x_{j}\\\sum _{j=1}^{n}a_{2j}x_{j}\\\vdots \\\sum _{j=1}^{n}a_{mj}x_{j}\end{pmatrix}}\,.}$

The ${\displaystyle {}i}$-th component of the image vector is

${\displaystyle {}y_{i}=\left(a_{i1},\,a_{i2},\,\ldots ,\,a_{in}\right){\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}=\sum _{j=1}^{n}a_{ij}x_{j}\,.}$

So one has to apply the ${\displaystyle {}i}$-th row of the matrix to the column vector ${\displaystyle {}x}$ in the described way.