Linear Algebra **Drawing Transition Diagrams for Markov Chains****Task:**Draw a transition diagram that corresponds to the following Markov chain:\[ \begin{pmatrix}a_{k+1} \\b_{k+1} \\c_{k+1} \end{pmatrix} = \begin{pmatrix}0.98 & 0.80 & 0.95 \\0.02 & 0 & 0 \\0 & 0.20 & 0.05 \end{pmatrix}\begin{pmatrix}a_k \\b_k \\c_k \end{pmatrix} \]**Explanation:**In this task, we are given a matrix that describes the transition probabilities of a Markov chain. The vector $\begin{pmatrix} a_{k+1} \\ b_{k+1} \\ c_{k+1} \end{pmatrix}$ represents the state of the system at the next time step, while the vector $\begin{pmatrix} a_k \\ b_k \\ c_k \end{pmatrix}$ represents the current state of the system. The matrix in between these vectors describes the probability of transitioning from one state to another.To create the transition diagram:1. **Identify the States:** - $a_k$ - $b_k$ - $c_k$2. **Draw Nodes for Each State:** - Represent each state $a_k$, $b_k$, and $c_k$ as a circle with respective labels.3. **Draw Directed Arrows According to Transition Probabilities:** - From $a_k$ to $a_{k+1}$ with weight 0.98 - From $a_k$ to $b_{k+1}$ with weight 0.80 - From $a_k$ to $c_{k+1}$ with weight 0.95 - From $b_k$ to $a_{k+1}$ with weight 0.02 - From $b_k$ to $c_{k+1}$ with weight 0.20 - From $c_k$ to $a_{k+1}$ with weight 0.05 4. **Label

From the given information, there are three states: ak, bk, ck. And the transition matrix is,

Answered: Draw a transition diagram that…

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

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Linear Algebra

**Drawing Transition Diagrams for Markov Chains**

**Task:**
Draw a transition diagram that corresponds to the following Markov chain:

\[
\begin{pmatrix}
a_{k+1} \\
b_{k+1} \\
c_{k+1}
\end{pmatrix}
=
\begin{pmatrix}
0.98 & 0.80 & 0.95 \\
0.02 & 0 & 0 \\
0 & 0.20 & 0.05
\end{pmatrix}
\begin{pmatrix}
a_k \\
b_k \\
c_k
\end{pmatrix}
\]

**Explanation:**

In this task, we are given a matrix that describes the transition probabilities of a Markov chain. The vector $\begin{pmatrix} a_{k+1} \\ b_{k+1} \\ c_{k+1} \end{pmatrix}$ represents the state of the system at the next time step, while the vector $\begin{pmatrix} a_k \\ b_k \\ c_k \end{pmatrix}$ represents the current state of the system. The matrix in between these vectors describes the probability of transitioning from one state to another.

To create the transition diagram:

1. **Identify the States:**
- $a_k$
- $b_k$
- $c_k$

2. **Draw Nodes for Each State:**
- Represent each state $a_k$, $b_k$, and $c_k$ as a circle with respective labels.

3. **Draw Directed Arrows According to Transition Probabilities:**
- From $a_k$ to $a_{k+1}$ with weight 0.98
- From $a_k$ to $b_{k+1}$ with weight 0.80
- From $a_k$ to $c_{k+1}$ with weight 0.95
- From $b_k$ to $a_{k+1}$ with weight 0.02
- From $b_k$ to $c_{k+1}$ with weight 0.20
- From $c_k$ to $a_{k+1}$ with weight 0.05

4. **Label

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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