Applications: Projects requiring solutions of systems

Solving systems of equations

Nodal analysis of circuits - Uses systems of equations to find the current through each loop of a circuit including batteries and resisters.  Nodal analysis creates linear equations using Kirchhoff's Laws of junctions and paths.  This is a popular project for students who have studied some physics.  One value of this project is the ability to create overdetermined, consistent systems of equations, which helps students understand rows of zeros in the RREF form of augmented matrices.  This article on Nodal Analysis of Electric Circuits has a clear explanation.

Loop analysis of circuits - Uses systems of equations to find the current through each loop of a circuit including batteries and resisters.  Loop analysis creates linear equations using Kirchhoff's Laws of loops.  Again, a popular project for students who have studied some physics and also has the opportunity for overdetermined, consistent systems.  Equivalent in results to nodal analysis, this could be combined or assigned separately.  This article on Loop Analysis of Electric Circuits has a clear explanation.

Curve fitting - Using systems of equations a student finds the coefficients of a polynomial of degree n - 1 to fit n points.  I don't think of this as a juicy application that gives the student an appreciation for how linear algebra is used in the world.  Fitting an n-degree polynomial to m points using least squares or other methods is more likely to happen.

Applications: Projects using vector spaces

Inner product spaces

Curve fitting using least squares - Uses matrix multiplication, inverses, and equations to find coefficients of a curve that fits a set of points.  I have three misgivings about curve fitting with least squares: it can be done without any understanding, it is not representative of least squares problems in general, and the various spaces involved confuses the issue.  Taking the last point first, the problem involves points in 2D or 3D space, matrices in n x k space where n is the number of points and k the terms of the curve being fit, and coefficients that live in k-space.  If we are trying to fit a line, the coefficients are 2D, but that 2D space is not the 2D space of the original points.

If a student is assigned this project without have learned about projections, they can do the calculations anyway, since they just require matrix multiplication and solving systems of equations.  The process of setting up the matrices does not promote deeper understanding of inner product spaces, and so if the student is going to fit curves, they might as well use Excel, which also doesn't deepen their understanding of linear algebra.

Finally, if a student learns least squares in this way, then they have difficulty transferring this concept to the solutions of noisy systems using least squares and don't think of least squares as a method of approximating solutions, but rather of fitting curves.