Linear mapping/Matrix/Relation/Section

Due to fact, a linear mapping

The effect of several linear mappings from ${}\mathbb {R} ^{2}$ to itself, represented on a brain cell.

$\varphi \colon K^{n}\longrightarrow K^{m}$

is determined by the images ${}\varphi (e_{j})$, ${}j=1,\ldots ,n$, of the standard vectors. Every ${}\varphi (e_{j})$ is a linear combination

${}\varphi (e_{j})=\sum _{i=1}^{m}a_{ij}e_{i}\,,$

and therefore the linear mapping is determined by the elements ${}a_{ij}$. So, such a linear map is determined by the ${}mn$ elements ${}a_{ij}$, ${}1\leq i\leq m$, ${}1\leq j\leq n$, from the field. We can write such a data set as a matrix. Because of the determination theorem, this holds for linear maps in general, as soon as in both vector spaces bases are fixed.

Definition

Let ${}K$ denote a field, and let ${}V$ be an ${}n$-dimensional vector space with a basis ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$, and let ${}W$ be an ${}m$-dimensional vector space with a basis ${}{\mathfrak {w}}=w_{1},\ldots ,w_{m}$.

For a linear mapping

$\varphi \colon V\longrightarrow W,$

the matrix

${}M=M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )=(a_{ij})_{ij}\,,$

where ${}a_{ij}$ is the ${}i$-th coordinate of ${}\varphi (v_{j})$ with respect to the basis ${}{\mathfrak {w}}$, is called the describing matrix for ${}\varphi$ with respect to the bases.

For a matrix ${}M=(a_{ij})_{ij}\in \operatorname {Mat} _{m\times n}(K)$, the linear mapping ${}\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M)$ determined by

$v_{j}\longmapsto \sum _{i=1}^{m}a_{ij}w_{i}$

in the sense of fact,

is called the linear mapping determined by the matrix ${}M$.

For a linear mapping ${}\varphi \colon K^{n}\rightarrow K^{m}$, we always assume that everything is with respect to the standard bases, unless otherwise stated. For a linear mapping ${}\varphi \colon V\rightarrow V$ from a vector space in itself (what is called an endomorphism), one usually takes the same bases on both sides. The identity on a vector space of dimension ${}n$ is described by the identity matrix, with respect to every basis.

Theorem

Let ${}K$ be a field, and let ${}V$ be an ${}n$-dimensional vector space with a basis ${}{\mathfrak {v}}=v_{1},\ldots ,v_{n}$, and let ${}W$ be an ${}m$-dimensional vector space with a basis ${}{\mathfrak {w}}=w_{1},\ldots ,w_{m}$. Then the mappings

$\varphi \longmapsto M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ){\text{ and }}M\longmapsto \varphi _{\mathfrak {w}}^{\mathfrak {v}}(M),$

defined in definition, are inverse to each other.

Proof

This proof was not presented in the lecture.

$\Box$

Example

A linear mapping

$\varphi \colon K^{n}\longrightarrow K^{m}$

is usually described by the matrix ${}M$ with respect to the standard bases on the left and on the right. The result of the matrix multiplication

${}{\begin{pmatrix}y_{1}\\\vdots \\y_{m}\end{pmatrix}}=M{\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}\,$

can be interpreted directly as a point in ${}K^{m}$. The ${}j$-th column of ${}M$ is the image of the ${}j$-th standard vector ${}e_{j}$.