Mapping/Linear algebra/Section

We discuss two classes of mappings which are in the framework of linear algebra very important. They are both so-called linear mappings.

Example

Let ${}a\in \mathbb {R}$ be fixed. This real number defines a mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax.$

For ${}a=0$, this is the constant zero mapping. For ${}a\neq 0$, we have a bijective mapping, the inverse mappiung is

$y\longrightarrow {\frac {1}{a}}y.$

Here, the inverse mapping has a similar form as the mapping itself.

Example

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

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

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

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

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

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

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

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

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

It is a goal of linear algebra to determine, in dependence of the entries ${}a_{ij}$, whether the mapping defined by the matrix is injective, surjective or bijective, and how, in the bijective case, the inverse mapping looks like.

Example

A healthy breakfast starts with a fruit salad. The following table shows how much vitamin C, calcium and magnesium various fruits have (in milligram with respect to 100 gram of the fruit).

	apple	orange	grapes	banana
vitamin C	${}12$	${}53$	${}4$	${}9$
calcium	${}7$	${}40$	${}12$	${}5$
magnesium	${}6$	${}10$	${}8$	${}27$

This table yields a mapping, which assigns to a ${}4$-tuple ${}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}$, representing the used fruits, the content of the resulting fruit salad with respect to vitamin C, calcium and magnesium in the form of a ${}3$-tuplel ${}{\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}$. This mapping can be described with the matrix

${\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}$

using matrix multiplication as

${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}\longmapsto {\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}={\begin{pmatrix}12x_{1}+53x_{2}+4x_{3}+9x_{4}\\7x_{1}+40x_{2}+12x_{3}+5x_{4}\\6x_{1}+10x_{2}+8x_{3}+27x_{4}\end{pmatrix}}={\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}.$