$v_2 = A v_1 = \begin{bmatrix} 1/4 \ 1/2 \ 1/4 \end{bmatrix}$

$v_1 = A v_0 = \begin{bmatrix} 1/6 \ 1/2 \ 1/3 \end{bmatrix}$

Let's say we have a set of $n$ web pages, and we want to compute the PageRank scores. We can create a matrix $A$ of size $n \times n$, where the entry $a_{ij}$ represents the probability of transitioning from page $j$ to page $i$. If page $j$ has a hyperlink to page $i$, then $a_{ij} = \frac{1}{d_j}$, where $d_j$ is the number of hyperlinks on page $j$. If page $j$ does not have a hyperlink to page $i$, then $a_{ij} = 0$.

Imagine you're searching for information on the internet, and you want to find the most relevant web pages related to a specific topic. Google's PageRank algorithm uses Linear Algebra to solve this problem.