Example of stochastic matrix of mapping

In order to understand the notion of stochastic matrix associatedto a mapping and its dual, we will work through a simple example.Let $X=\\{a,b,c\\}$ and let $Y=\\{d,e\\}$ , and define the mapping $f\\colon X\\to Y$ as follows:

	$\\displaystyle f(a)$	$\\displaystyle=d$
	$\\displaystyle f(b)$	$\\displaystyle=d$
	$\\displaystyle f(c)$	$\\displaystyle=e$

Then ${\\mathcal{V}}X$ is a 3-dimensional real vector space with basis

\\delta_{a}=\\begin{pmatrix}1\\cr 0\\cr 0\\end{pmatrix},\\qquad\\delta_{b}=\\begin{%pmatrix}0\\cr 1\\cr 0\\end{pmatrix},\\qquad\\delta_{c}=\\begin{pmatrix}0\\cr 0\\cr 1%\\end{pmatrix}

and ${\\mathcal{V}}Y$ is a 3-dimensional real vector space with basis

\\delta_{c}=\\begin{pmatrix}1\\cr 0\\end{pmatrix},\\qquad\\delta_{d}=\\begin{pmatrix}%0\\cr 1\\end{pmatrix}

and

{{\\mathcal{V}}f}=\\begin{pmatrix}1&1&0\\cr 0&0&1\\end{pmatrix}.

To form the dual, we first renormalize the rows to sum to unity,then transpose:

\\begin{pmatrix}1&1&0\\cr 0&0&1\\end{pmatrix}\\xrightarrow{ren}\\begin{pmatrix}%\\frac{1}{2}&\\frac{1}{2}&0\\cr 0&0&1\\end{pmatrix}\\xrightarrow{*}\\begin{pmatrix}%\\frac{1}{2}&0\\cr\\frac{1}{2}&0\\cr 0&1\\end{pmatrix}

Next, to illustrate inclusions, we shall examine the map $i\\colon Y\\hookrightarrow X$ defined as follows:

	$\\displaystyle f(d)$	$\\displaystyle=$	$\\displaystyle a$
	$\\displaystyle f(e)$	$\\displaystyle=$	$\\displaystyle b$

Following the same procedures as above, for this map we find that

{{\\mathcal{V}}i}=\\begin{pmatrix}1&0\\\\0&1\\\\0&0\\end{pmatrix}

and

({{\\mathcal{V}}i})^{\atural}=\\begin{pmatrix}1&0&0\\\\0&1&0\\end{pmatrix}