Time:  TuTh 11:0012:15  Location:  Mudd 337 
Instructor:  Marc
Spiegelman (home page) 
Office Hours:  Tues/Thurs. 1:452:45 211 Mudd (APAM) 
TAs:  Anthony Zacharakis 
Office Hours:  Office hours in Mudd 292: Mondays 11:00am12:30 
Text:  Gilbert Strang, Introduction to Linear
Algebra, 3rd Ed. 
Grading:  Homework 20%, Quizs 40% Final 40% 


Study
Guide 2007
v1.3 (The Long Version 2004 v1.5) Other Study resources: Strang site at MIT 

Week  Date  Chapter  Subject  Problem Set 

1  2 Sep  1.11.2  Introduction to scalars and Vectors; definitions, vectors in
2D, vector addition and scalar multiplications, vectors in 3D and
nD. Linear Combinations of
Vectors, The dot product, length of a vector and "unit" vectors.
Here is a
simple matlab script for messing
with
vectors (to download use the "save link as" function) PDF Note Template for Lecture 01 
1 
4 Sep 
2.12.2  Vectors and linear equations, matrices, the primacy of Ax=b.
Solving linear Equations 1: the idea of elimination. Here is a matlab script for Ax=b and
matlab functions for plotting 2D
and 3D vectors. PDF Note Templates for Lecture 02 

2  9 Sep 
2.32.4  Solving linear Equations 2: Elimination using matrices. the
matrices E and P. Rules for matrix operations, block matrices PDF Note Templates for Lecture 03 
2 
11 Sep 
2.5  Solving linear Equations 3: Inverse Matrices: properties of
the inverse. Solution by GaussJordan Elimination. Singular matrices,
Inverse of L PDF Note Templates for Lecture 04 

3  16 Sep 
2.62.7  Solving Linear equations 4: A inverse contd., Elimination =
Factorization A=LU. PDF Note Templates for Lecture 05 
3 
18 Sep  Solving Linear Equations 5: Final comments on the LU:
row swaps, permutation matrices and PA=LU. The LU in action: Matlab Functions for the SpringDemo. Another example: the "Fish Farm Problem": a simplified example of solutions by LU of large sparse matrices. A matlab script for solving large tridiagonal systems such as the "Fish Farm" problem. Plus a directory of example matlab functions for doing the same. Symmetric Matrices A^{T}=A. PDF Note Templates for Lecture 06 

4  23 Sep 
3.13.2  Vector Spaces and Subspaces:
definitions and properties. Column space and the Nullspace. PDF Note Templates for Lecture 07 
4 
25 Sep  3.23.3  The Null space contd. Rank and the reduced row echelon form.
Solutions of Ax=0. Special solutions. PDF Note Templates for Lecture 08 

5  30 Sep 
343.6  The complete solution
to Ax=b,
Linear Independence, Span, Basis and
Dimension, PDF Note Templates for Lecture 09 
5 
2 Oct  3.64.1  Dimensions of the four
subspaces (Column Space, Row Space, Null Space and Left Null Space) Orthogonality of the four subspaces and The Big Picture PDF Note Templates for Lecture 10 

6  7 Oct 
Quiz
#1: Ax=b redux...Gaussian Elimination, Linear systems,GaussJordan Elimination and A^{1}, PA=LU,Vector Spaces and Subspaces through C(A) and N(A) (problem set 4). Quiz 1 Histogram 

9 Oct  4.2  Projections 1: projection
onto a
line, projection onto subspaces PDF Note Templates for Lecture 11 
6 

7  14 Oct 
4.3  Projections 2: Least Squares
Approximations. And here is an
example Matlab
Script to calculate
best fit lines, quadratics and cubics together with some optional data.
Save
the script and data to a directory (use Save Link as) and then start
matlab from there. Just type the name of the matlab script
"LeastSquaresFun" at the matlab prompt and it should do the rest. PDF Note Templates for Lecture 12 

16 Oct 
4.34.4  Projections 3: Least Squares
and Orthogonal
Bases. Q matrices examples
(projection, rotation, reflection) and
properties. A>Q by GramSchmidt orthogonalization. PDF Note Templates for Lecture 13 

8  21 Oct  4.4  The QR decomposition and
Least Squares PDF Note Templates for Lecture 14 
7 
23 Oct  5.1  Determinants:
Properties,
Formulas, applications PDF Note Templates for Lecture 15 

9  4 Nov  Election
Day! Go Vote! Make History! (History was Made!) 

6 Nov 
6.1  Eigensystems 1: Introduction
to eigenvalues and eigenvectors PDF Note Templates for Lecture 16 
8 

10  11 Nov  6.2 
Eigensystems 2: Diagonalizing a matrix. Matrix powers A^{n }PDF Note Templates for Lecture 17  
13 Nov Last day to drop for SEAS 
Quiz #2: The four subspaces of the apocalypse. Projections, LeastSquares problems, Q matrices, and GramSchmidt Quiz 2 Histogram 

14 Nov Optional Bonus lecture 
6.2  Eigensystems 3: Matrix Powers
cont'd...the power method and the google algorithm A nice presentation on LinkAnalysis by Amy Langville. A Matlab Demo illustrating the 6 node example from Langville. Figures showing the evolution of the pagerank vector and convergence of the power method for this example. PDF Note Templates for Lecture 18 

11  18 Nov  6.3  Eigensystems 4: Applications
to
systems of ODE's 
linear dynamical systems A nice phaseportrait generator at MIT A web site tutorial for simple nonlinear dynamical systems (for the real stuff see DSWEB) PDF Note Templates for Lecture 19 
9 
20 Nov  6.46.5  Eigensystems 5
similar
matrices, symmetric matrices, and
positive definite
matrices PDF Note Templates for Lecture 20 

12  25 Nov  6.7, 7.4  Eigensystems 6: The
fabulous
SVD (singular Value
Decomposition) PDF Note Templates for Lecture 21 

27 Nov  Tofu Turkey Day!  
13  2 Dec  The
SVD of a
singular matrix, The big picture, the
pseudoinverse. Applications 1: leastsquares and Total Least Squares. Some notes from Eero Simoncelli on general leastsquares optimization. Some Matlab code for TLS. More matlab code for the demos I showed in class PDF Note Templates for Lecture 22 

4 Dec  Applications 2:
Applications of the SVD. Here's a bit of
image processing fun using the SVD. Save these two links to a Matlab
Script and this Matlab Image
then run. Here is a PDF
file of the presentation I gave in
class which has a bit more of an explanation of the technique. PDF Note Templates for Lecture 23 

14  9 Dec  
Class Review Study Guide 2007 v1.3 PDF Note Templates for Lecture 24 & The future (PDE's and Scientific computation) 

Final
Exam: 16 December 