E3101: Applied Math  I

Introduction to Linear Algebra

Time: Tu-Th 11:00-12:15 Location: Mudd 337
Instructor: Marc Spiegelman   (home page)
Office Hours: Tues/Thurs. 1:45-2:45
 211 Mudd (APAM)
TAs: Anthony Zacharakis
Office Hours:  Office hours in Mudd 292:
Mondays 11:00am-12:30
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 1.1-1.2 Introduction to scalars and Vectors; definitions, vectors in 2-D, vector addition and scalar multiplications, vectors in 3-D and n-D. 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
4 Sep
2.1-2.2 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 3-D vectors.
PDF Note Templates for Lecture 02
2 9 Sep
2.3-2.4 Solving linear Equations 2: Elimination using matrices. the matrices E and P. Rules for matrix operations, block matrices
PDF Note Templates for Lecture 03
11 Sep
2.5 Solving linear Equations 3: Inverse Matrices: properties of the inverse. Solution by Gauss-Jordan Elimination. Singular matrices, Inverse of L
PDF Note Templates for Lecture 04
3 16 Sep
2.6-2.7 Solving Linear equations 4: A inverse contd., Elimination = Factorization A=LU.
PDF Note Templates for Lecture 05
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 AT=A.
PDF Note Templates for Lecture 06
4 23 Sep
3.1-3.2 Vector Spaces and Subspaces: definitions and properties. Column space and the Nullspace.
PDF Note Templates for Lecture 07
25 Sep 3.2-3.3 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
3-4-3.6 The complete solution to Ax=b, Linear Independence, Span, Basis and Dimension,
PDF Note Templates for Lecture 09

2 Oct 3.6-4.1 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,Gauss-Jordan 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


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
4.3-4.4 Projections 3: Least Squares and Orthogonal Bases. Q matrices examples (projection, rotation, reflection) and properties. A->Q by Gram-Schmidt orthogonalization.
PDF Note Templates for Lecture 13
8 21 Oct 4.4 The QR decomposition and Least Squares
PDF Note Templates for Lecture 14
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
10 11 Nov 6.2
Eigensystems 2: Diagonalizing a matrix. Matrix powers An
PDF Note Templates for Lecture 17
13 Nov
Last day to drop for SEAS
Quiz #2:
The four subspaces of the apocalypse. Projections, Least-Squares problems, Q matrices, and Gram-Schmidt
Quiz 2 Histogram
14 Nov
Optional Bonus
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 phase-portrait generator at MIT
A web site tutorial for simple non-linear dynamical systems (for the real stuff see  DSWEB)
PDF Note Templates for Lecture 19
20 Nov 6.4-6.5 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 pseudo-inverse.  Applications 1: least-squares and Total Least Squares.
Some notes from Eero Simoncelli on general least-squares 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

marc spiegelman
Last modified: 01 Sept 2008