Thursday, June 1, 2017

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.