Linear mapping/Matrix/Relation/2/Section

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

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

and hence determined by the elements ${}a_{ij}$. This means all together that such a linear mapping is given by the ${}mn$ elements ${}a_{ij}$, ${}1\leq i\leq m$, ${}1\leq j\leq n$. Such a set of data can be written as a matrix. Due to fact, this observation holds for all finite-dimensional vector spaces, as long as bases are fixed on the source space and on the target space of the linear mapping.

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.

If ${}V=W$, then we are usually interested in the describing matrix with respect to one basis ${}{\mathfrak {v}}$ of ${}V$.

Example

Let ${}V$ denote a vector space with bases ${}{\mathfrak {v}}$ and ${}{\mathfrak {w}}$. If we consider the identity

$\operatorname {Id} \colon V\longrightarrow V$

with respect to the basis ${}{\mathfrak {v}}$ on the source and the basis ${}{\mathfrak {w}}$ on the target, we get, because of

${}\operatorname {Id} (v_{j})=v_{j}=\sum a_{ij}w_{i}\,,$

directly

${}M_{\mathfrak {w}}^{\mathfrak {v}}(\operatorname {Id} )=M_{\mathfrak {w}}^{\mathfrak {v}}\,.$

This means that the describing matrix of the identical linear mapping is the transformation matrix for the base change from ${}{\mathfrak {v}}$ to ${}{\mathfrak {w}}$.

Lemma

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

$\Psi _{\mathfrak {v}}\colon K^{n}\longrightarrow V$

and

$\Psi _{\mathfrak {w}}\colon K^{m}\longrightarrow W$

be the corresponding mappings. Let

$\varphi \colon V\longrightarrow W$

denote a linear mapping with describing matrix ${}M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )$. Then

${}\varphi \circ \Psi _{\mathfrak {v}}=\Psi _{\mathfrak {w}}\circ M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )\,$

hold, that is, the diagram

${\begin{matrix}K^{n}&{\stackrel {\Psi _{\mathfrak {v}}}{\longrightarrow }}&V&\\\!\!\!\!\!M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )\downarrow &&\downarrow \varphi \!\!\!\!\!&\\K^{m}&{\stackrel {\Psi _{\mathfrak {w}}}{\longrightarrow }}&W&\!\!\!\!\!\\\end{matrix}}$

commutes. For a vector ${}v\in V$,

we can compute ${}\varphi (v)$ by determining the coefficient tuple of ${}v$ with respect to the basis ${}{\mathfrak {v}}$, applying the matrix ${}M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )$ and determining for the resulting ${}m$-tuple the corresponding vector with respect to ${}{\mathfrak {w}}$.

Proof

See exercise.

$\Box$

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

We show that both compositions are the identity. We start with a matrix ${}M={\left(a_{ij}\right)}_{ij}$ and consider the matrix

$M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M)).$

Two matrices are equal, when the entries coincide for every index pair ${}(i,j)$. We have

${}{\begin{aligned}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M)))_{ij}&=i-{\text{th coordinate of }}(\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M))(v_{j})\\&=i-{\text{th coordinate of }}\sum _{i=1}^{m}a_{ij}w_{i}\\&=a_{ij}.\end{aligned}}$

Now, let ${}\varphi$ be a linear mapping, we consider

$\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )).$

Two linear mappings coincide, due to fact, when they have the same values on the basis ${}v_{1},\ldots ,v_{n}$. We have

${}(\varphi _{\mathfrak {w}}^{\mathfrak {v}}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi )))(v_{j})=\sum _{i=1}^{m}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))_{ij}\,w_{i}\,.$

Due to the definition, the coefficient ${}(M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ))_{ij}$ is the ${}i$-th coordinate of ${}\varphi (v_{j})$ with respect to the basis ${}w_{1},\ldots ,w_{m}$. Hence, this sum equals ${}\varphi (v_{j})$.

$\Box$

We denote the set of all linear mappings from ${}V$ to ${}W$ by ${}\operatorname {Hom} _{K}{\left(V,W\right)}$. fact means that the mapping

$\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Mat} _{m\times n}(K),\varphi \longmapsto M_{\mathfrak {w}}^{\mathfrak {v}}(\varphi ),$

is bijective with the given inverse mapping. A linear mapping

$\varphi \colon V\longrightarrow V$

is called an endomorphism. The set of all endomorphisms on ${}V$ is denoted by ${}\operatorname {End} _{K}{\left(V\right)}$.