Need help with the Linear Algebra problem. The images are the examples. Use a weighted Colley Method which ranks games won (loss) early in the season as 0.5 wins (losses), games won in the mid-season ranked as 1 win (loss), and games won late in the season as 2 wins (losses) to rank the following system: During the Early Part of the Season: Team B beats C once. Team D beats E once and C once. Team E beats B once.During the Middle Part of the Season: Team A beats C once. Team D beats B once. Team E beats C onceDuring the Later Part of the Season: Team A beats E once and C once. Team E beats B once.a. Draw a directed graph of this season.b. Write the system of equations for this season using the Massey method.c. Solve the system (using least squares) to find the ratings for each team, and use those ratings to rank them. Example 3.1. if Team A beat Team B, Team C beat Team A, Team C beat team B, and TeamB beat Team C, using the Colley Method we would have the following system of equations:(2+2)rA = 1 +(2+3)rB = 1 +(2+3)rc = 1 +Rearranging the system gives us:Solving this system gives us1 121-2Rank2232214rArBrc = 1―rA +5rB - 2rc = 0.5-TA - 2rB+5rc = 1.5This gives us a nice symmetric matrix system, Cr = b:4 −1 -1-1 5 -2−2-1525+rB+rc+rA +2rcTeamсAB+rA +2rBΤΑTBrc0.5ΤΑTB0.42857 . We order the team rankings by the value0.57143rcof the ratings with the largest rating corresponding with the top ranked team. Putting theratings in order from largest to smallest, we get the following ranking: Team C first, TeamA second, and Team B third.=10.51.5(6) 3.2 Lab 3: In class portion3.2.1 The Colley MethodAs part of his mathematics PhD dissertation, Dr. Wesley Colley developed a method to rank teamsin a way that incorporates strength of schedule which can make it a little more accurate than winpercentage. At the start of the season, a team's winning percentage would 0/0 which is undefined!The Colley Method says an untried team should have a rating of 50%, 50-50 chance the team willwin or lose. Colley also wants to have the average of all rating be roughly 0.5 as well. So we adjusthow we calculate our win percentage.Wi9Regular Winning Percentage is calculated by where w is the total number of wins fortiTeam i and t; is the total number of games placed by Team i.Wi + 1The Colley Winning Percentage is calculated asSince, after each game, the denom-ti + 2inator increases by 1, the losers rating will decreasing and the winners rating will increaseafter each game.To set up his system, Colley sets the rating for each team, r¡, equal to its Colley winWi + 1percentage. That is, r; =In order to incorporate strength of schedule, Colley doest₁ + 2the following algebraic manipulation (note l, denotes the total number of losses for Team i):(1)(2)(3)0. Rearranging our equation, we nowNote we use the fact that wiget:(2 + ti) ri2+2=Pi =(2+ti)ri = 1+ Wi-Wi + 1ti + 2Wi Wi lj1+ + +2 2 2and 1/2-1/2½-½(2+tį)rį = 1 +Wi lį2=(2+ti)ri = 1 ++24li(4)Wi + liNote thatis just the total number of games played by team i divided by 2. This is2where Colley made an adjustment to his system in order to incorporate strength of schedule.Wi + ljHe does this by replacing with the sum of the ratings of the teams played by team i.2Thus our new Colley system has the following set up. For each Team i, we get the followingequation:2Wi+li2Wi ·lį2where S is the sum of the ratings of teams played by Team i.+ S,(5)

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Need help with the Linear Algebra problem. The images are the examples.

Use a weighted Colley Method which ranks games won (loss) early in the season as 0.5 wins (losses), games won in the mid-season ranked as 1 win (loss), and games won late in the season as 2 wins (losses) to rank the following system:

During the Early Part of the Season: Team B beats C once. Team D beats E once and C once. Team E beats B once.

During the Middle Part of the Season: Team A beats C once. Team D beats B once. Team E beats C once

During the Later Part of the Season: Team A beats E once and C once. Team E beats B once.

a. Draw a directed graph of this season.

b. Write the system of equations for this season using the Massey method.

c. Solve the system (using least squares) to find the ratings for each team, and use those ratings to rank them.

Example 3.1. if Team A beat Team B, Team C beat Team A, Team C beat team B, and Team
B beat Team C, using the Colley Method we would have the following system of equations:
(2+2)rA = 1 +
(2+3)rB = 1 +
(2+3)rc = 1 +
Rearranging the system gives us:
Solving this system gives us
1 1
2
1-2
Rank
2
2
3
2
2
1
4rArBrc = 1
―rA +5rB - 2rc = 0.5
-TA - 2rB+5rc = 1.5
This gives us a nice symmetric matrix system, Cr = b:
4 −1 -1
-1 5 -2
−2
-1
5
25
+rB+rc
+rA +2rc
Team
с
A
B
+rA +2rB
ΤΑ
TB
rc
0.5
ΤΑ
TB
0.42857 . We order the team rankings by the value
0.57143
rc
of the ratings with the largest rating corresponding with the top ranked team. Putting the
ratings in order from largest to smallest, we get the following ranking: Team C first, Team
A second, and Team B third.
=
1
0.5
1.5
(6)

3.2 Lab 3: In class portion
3.2.1 The Colley Method
As part of his mathematics PhD dissertation, Dr. Wesley Colley developed a method to rank teams
in a way that incorporates strength of schedule which can make it a little more accurate than win
percentage. At the start of the season, a team's winning percentage would 0/0 which is undefined!
The Colley Method says an untried team should have a rating of 50%, 50-50 chance the team will
win or lose. Colley also wants to have the average of all rating be roughly 0.5 as well. So we adjust
how we calculate our win percentage.
Wi
9
Regular Winning Percentage is calculated by where w is the total number of wins for
ti
Team i and t; is the total number of games placed by Team i.
Wi + 1
The Colley Winning Percentage is calculated as
Since, after each game, the denom-
ti + 2
inator increases by 1, the losers rating will decreasing and the winners rating will increase
after each game.
To set up his system, Colley sets the rating for each team, r¡, equal to its Colley win
Wi + 1
percentage. That is, r; =
In order to incorporate strength of schedule, Colley does
t₁ + 2
the following algebraic manipulation (note l, denotes the total number of losses for Team i):
(1)
(2)
(3)
0. Rearranging our equation, we now
Note we use the fact that wi
get:
(2 + ti) ri
2+2
=
Pi =
(2+ti)ri = 1+ Wi
-
Wi + 1
ti + 2
Wi Wi lj
1+ + +
2 2 2
and 1/2-1/2
½-½
(2+tį)rį = 1 +
Wi lį
2
=
(2+ti)ri = 1 +
+
24
li
(4)
Wi + li
Note that
is just the total number of games played by team i divided by 2. This is
2
where Colley made an adjustment to his system in order to incorporate strength of schedule.
Wi + lj
He does this by replacing with the sum of the ratings of the teams played by team i.
2
Thus our new Colley system has the following set up. For each Team i, we get the following
equation:
2
Wi+li
2
Wi ·lį
2
where S is the sum of the ratings of teams played by Team i.
+ S,
(5)

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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