An approximate syllabus for Math 401                            Professor Dancis

The textbook is my book entitled:  An Introduction to Linear Algebra for Science and 
Engineering Students.  This book will be available from the book store in the student union.

Chapter  3. Train students to become fluent in symbolic matrix algebra calculations, Remove 
misconceptions.
Train students to do proof-by-calculations.
The connections between Symbolic Matrix Algebra and Matrix transformations
Applications: Superposition for (electrical) D.C. resistance circuits. 
Finding particular solutions to special Systems of Non-homogeneous Linear Differential 
Equations.
Perpendicular projections and Least square fits (including linear and multiple regressions) 

Chapter  4    	Systems of linear algebraic Equations  (Assuming students know Gauss)
Perturbation Theory - the surprising effects of measurement errors (for matrix  M  and vector  w) 
on the solution  v,  to  Mv=w.  Also how to quickly obtain good estimates.		
Design problems for  Mv=w  (How tight must the tolerances on  w  be, in order that the 
specifications on  v,  will be satisfied?)
Solving Word Problems (Translating word problems into equations.)
Appendix	Computerized Tomography, the idea behind CT scanners
The Bouguer-Lambert Law of Photometry.

Chapter  2    General Linear equations 	
Easily recognizing general linear transformations
Solving general linear equations with applications to linear differential equations and linear 
difference equations (all with constant coefficients and no double roots.)  No previous 
knowledge of linear differential equations is required.  The emphasis will be explaining Math 
246.
Eigenfunctions 
The Separation of Variables Method for solving homogeneous linear Partial Differential 
Equations. -- Applied to the Heat Diffusion Equation, the Wave Equation and the steady-state 
Heat Equation.
Uniqueness and Stability of Solutions for Heat Diffusion Equations.
Word Problems and Exponential decay to steady state

Chapter  5    	Background for Eigenvalue Theory  
Non-standard linear coordinate systems for Rn.  
Graph equations like  (y-x) = (y+3x)5  using a change-of-basis matrix.

Chapter  6    	Eigenvalue Theory (without Jordan form -- no double roots)
Eigenvalues of special matrices.
Diagonalization  (and matrix similarity as an equivalence relation)
Solving finite difference equations:  vn+1 = M vn       Application to Arms Races.
The basic systems of linear differential equations  (v' = Mv)
including using a change-of-basis matrix to graph solutions and classifying theorems.

Gershgorin disc theory for estimating eigenvalues
With proof of Perron Theorem for diagonalizable stochastic/probability matrices.

Chapter  7 Quadratic Forms - Max and Min
The Spectral Theorem for real symmetric matrices

Chapter  8  	Positive Definite Matrices 
Theorem.   If  A  and  B  are positive definite nxn-matrices, then  AB  is diagonalizable with 
positive eigenvalues.  
Applications to Systems of second-order homogeneous linear differential equations arising in 
engineering -- including systems of springs and blocks, linear electrical circuits and diffusion.   
Also matrix formulas for the energy of systems of linear springs and blocks.