Linear Systems of Algebraic Equations

This page presents some topics from Linear Algebra needed for construction of solutions to systems of linear algebraic equations and some applications. We use matrices and vectors as essential elements in obtaining and expressing the solutions.

Markov Chains

Markov chaines could be absorbing or not absorbing.

Example. At a particular two-year college, a student has a probability of 0.25 of flunking out during a given year, a 0.15 probability of having to repeat the year, and 0.6 probability of finishing the year. There are four possible states: Freshman, Softmore, Has flunked out, and Graduate. We write a transition matrix in the form

\[ {\bf P} = \left( \begin{array}{c|c} {\bf R} & {\bf Q} \\ \hline {\bf 0} & {\bf I} \end{array} \right) = \left( \begin{array} {cccc} 0.15 & 0.6 & 0.25 & 0 \\ 0&0.15 & 0.25 & 0.6 \\ 0&0&1&0 \\ 0&0&0&1 \end{array} \right) . \]

sage: I=matrix(2,2,[1,0,0,1])
sage: Q2=matrix(QQ, 2,2,[0.15,0.6,0.0,0.15])
sage: Q2
[ 3/20 3/5 ]
[    0 3/20]
sage: F2=~(I-Q2)
sage: F2
[ 20/17 240/289]
[    0  20/17  ]
sage: R1=matrix(QQ, 2,2,[0.25,0.0,0.0.25,0.6])
sage: F2*R
[ 145/289 144/289 ]
[    5/17  12/17  ]