APPLICATIONS OF LINEAR ALGEBRA
Because of the ubiquity of vector spaces, linear algebra is used
in many fields of mathematics, natural sciences, computer science,
and social science. Below are just some examples of applications of
linear algebra.
Line of Best Fit
Scientists are often presented with a system that has no
solution and they must find an answer anyway. That is, they must
find a value that is as close as possible to being an answer.For
instance, suppose that we have a coin to use in flipping. This coin
has some proportionof heads to total flips, determined by how it is
physically constructed, and we want to know ifis near0.5. We can
get experimental data by flipping it many times. This is the result
a penny experiment, including some intermediate numbers.number of
flips 30 60 90
number of heads 16 34 51
Because of randomness, we do not find the exact proportion with
this sample there is no solution to this system.
That is, the vector of experimental data is not in the subspace
of solutions.
However, as described above, we want to find thethat most nearly
works. An orthogonal projection of the data vector into the line
subspace gives our best guess.
The estimate () is a bit high but not much, so probably the
penny is fair enough.The line with the slopeis called theline of
best fitfor this data.
Markov Chains
Here is a simple game: a player bets on coin tosses, a dollar
each time, and the game ends either when the player has no money
left or is up to five dollars. If the player starts with three
dollars, what is the chance that the game takes at least five
flips? Twenty-five flips?At any point, this player has either $0,
or $1, ..., or $5.We say that the player is in thestate,, ..., or.
A game consists of moving from state to state. For instance, a
player now in statehas on the next flip achance of moving to
stateand achance of moving to. The boundary states are a bit
different; once in stateor stat, the player never leaves.Letbe the
probability that the player is in stateafterflips. Then, for
instance, we have that the probability of being in stateafter
flipis. This matrix equation sumarizes.
With the initial condition that the player starts with three
dollars, calculation gives this.n=0n=1n=2n=3n=4n=24
As this computational exploration suggests, the game is not
likely to go on for long, with the player quickly ending in either
stateor state. For instance, after the fourth flip there is a
probability ofthat the game is already over.(Because a player who
enters either of the boundary states never leaves, they are said to
beabsorbing.)This game is an example of aMarkov chain, named for
A.A. Markov, who worked in the first half of the 1900's.Each vector
of's is aprobability vectorand the matrix is atransition matrix.The
notable feature of a Markov chain model is that it ishistorylessin
that with a fixed transition matrix, the next state depends only on
the current state, not on any prior states. Thus a player, say, who
arrives atby starting in state, then going to state, then to, and
then tohas at this point exactly the same chance of moving next to
stateas does a player whose history was to start in, then go to,
and to, and then to.
Video examples on Markov
chains:https://www.youtube.com/watch?v=nnssRe5DewEhttps://www.youtube.com/watch?v=uvYTGEZQTEs
