1.3 Applications of Systems of Linear Equations (Study Book 1.3

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1.3 Applicationsof Systems of Linear
Equations (Study Book 1.3 Larson 1.3)

• We can
• fit a polynomial to a set of given points
• set up solve systems of linear equations that
  arise from a network study either
• Larson Ex 5 (traffic flow)
• or Ex 6, 7 (electrical).

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• Fitting a polynomial
• Example
• Given 4 points with different (distinct) x-values
• eg (-2,3), (-1,5), (1,1), (2,4),
• we can always fit a polynomial of degree 3,
• a0 a1 x a2 x2 a3 x3 y
• with 4 as yet unknown coefficients ai
• to those points, precisely, as follows

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Substituting each of the 4 points into the
polynomial equation, gives 4 eqns in the 4
unknown coefficients a0 - 2 a1 4 a2 -
8 a3 3 a0 - a1 a2 - a3
5 a0 a1 a2 a3
1 a0 2 a1 4 a2 8 a3 4 The
augmented matrix is

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• Row-reducing gives
• The solutions are the values of the coefficients
  ai
• So the cubic polynomial that fits the points is
• Plot to check (eg using MATLAB).

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More Examples See Larson 1.3 Examples 1 - 4.

• (Note Ex 4 attempts to fit a poly to
• planetary data inappropriately.
• Fitting the poly to the logs, rather,
• leads to y2 x3, as Kepler found.)

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Network Analysis self-study.

• Networks arise in an attempt to analyse the
  interplay of inputs and outputs at junctions,
  and/or the flow along paths
• eg in electrical networks, economics, traffic
  analysis, computer engineering.
• Choose from Larson Ch 1.3
• study either Ex 5 (traffic flow)
• or Ex 6, 7 (electrical).

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Homework

• Larson Edwards Ch 1.3.
• Do Q 1, 5, 7, 11, 13.
• Do at least 2 network problems of your own
  choice, from Q 19-26 in Ed 4, or 21-28 in Ed 5.
• Write out full solutions to Q 5, 7, 11,
• 2 network problems.

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Objectives

• Know how to
• Fit a polynomial to a given set of points with
  distinct x-values
• Set up solve a system of linear equations for
  at least one type of network.

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• 1.4 Chapter Summary
• (Study Book Appendix A Larson end Ch1)
• An excellent summary follows in Appendix A
• in your Study Book
• Check the list at the end of Ch1 (Larson).
• Try some Review Exercises (Larson).
• Look at the Projects (Larson).