Tips on applications of matrices

[Kontilera]

Hello!
Im currently teaching a math course on a upper secondary school, where we are doing some linear algebra (it is the last math course before the students continue for university). After a brief introduction I want to show some interesting applications of matrices. Applications which can be explained in 40 min or less.

My ideas are:
Markov chains and Coupling matrices

Any more ideas? Maybe something regarding economics? Thanks!

 

[S.G. Janssens]

Any more ideas? Maybe something regarding economics?

For economics you could look up Leontief's input-output model. An introductory discussion can be found in D.C. Lay's Linear Algebra and its Applications. Recommended

 

[Kontilera]

Thanks! Will check it up!

 

[tygerdawg]

Since I'm a robot guy, my favorite is Homogeneous Coordinate Transformations. Might be heavy for high schoolers, but here's some random links from my collection for investigation.

http://mathworld.wolfram.com/EulerAngles.html

http://mathworld.wolfram.com/RotationMatrix.html

http://www.euclideanspace.com/maths/geometry/rotations/euler/

https://en.wikipedia.org/wiki/Transformation_matrix

 

[FactChecker]

Since I'm a robot guy, my favorite is Homogeneous Coordinate Transformations.

Along these lines, aircraft flight analysis and simulation is full of coordinate transformations (accelerometer box installation <=> airplane body <=> locally level <=> Earth Centered Earth Fixed; engine thrust <=> airplane body <=> relative wind; etc.)

 

[micromass]

Since I'm a robot guy, my favorite is Homogeneous Coordinate Transformations.

You mean projective geometry?

 

[nrqed]

Hello!
Im currently teaching a math course on a upper secondary school, where we are doing some linear algebra (it is the last math course before the students continue for university). After a brief introduction I want to show some interesting applications of matrices. Applications which can be explained in 40 min or less.

My ideas are:
Markov chains and Coupling matrices

Any more ideas? Maybe something regarding economics?Thanks!

Why not introduce matrices as observables in quantum mechanics and explain that their eigenvalues are the possible outcomes that a measurement of that observable may give?

 

[robphy]

https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis
http://www.photonics.byu.edu/ABCD_Matrix_tut.phtml

https://www.geogebra.org/m/VjhNaB8V linear transformations
http://www.cems.uvm.edu/~tlakoba/AppliedUGMath/other_Google/Wills.pdf (maybe just summarize the basic idea)

 

[hilbert2]

Linear algebra for traffic flow analysis: http://math.jbpub.com/advancedengineering/docs/Project8.2_TrafficFlow.pdf

 

[Devils]

z

Why not introduce matrices as observables in quantum mechanics and explain that their eigenvalues are the possible outcomes that a measurement of that observable may give?

Because they are high school students and not university students.

I have given Year 12 students assignments with question on Markov chains & Leontief models. They are expected to do some research themselves & we will discuss the questions in class (but not give exact solutions). These were at a Grammar school where expectation are high; I would not do it in a government school.

 

[FactChecker]

Why not introduce matrices as observables in quantum mechanics and explain that their eigenvalues are the possible outcomes that a measurement of that observable may give?

There are so many concrete applications of matrices that I think would be better.

 

[Dr. Courtney]

There are so many concrete applications of matrices that I think would be better.

Perhaps, but as one application among several, a simple explanation of matrices in quantum mechanics (like a two state system) can demonstrate the breadth of applications like few things can when limited to engineering and classical physics. I've never had the luxury of several 40 minute classes to discuss applications of matrices, but if I did, an example from quantum mechanics would be on the short list.

I usually consider the interests and perspective of the students when choosing examples. At the Air Force Academy, the math faculty worked hard to provide simple (yet not oversimplified) examples of Air Force and military applications. At the community college where I taught, students tended to be most receptive to examples from sports. Had a class with lots of aspiring medical professionals one time, so they tended to appreciate examples connected with health, anatomy, and medicine.

If I had a class where students showed an interest in physics, especially modern physics, I'd be eager to draw examples of applications from quantum mechanics. Of course, these are the students many math (and physics) teachers dream of and seldom see.

 

[Nidum]

The most common application of matrix methods is in solving the large sets of simultaneous equations that can arise when analysing physical problems using numerical methods .

Matrix methods have application in many real world problems like structural analysis and fluid mechanics .

One of the most commonly used tools in engineering is Finite Element Analysis . This is entirely based on matrix methods .

The solution of problems defined in matrix form is interesting in itself . Solution is possible by brute force but over the years many sophisticated methods have been developed to reduce computational time and improve accuracy .

.

 

[Kontilera]

thank you so much. I actually used several of the tips you gave me.