18.06 Linear Algebra

SES # TOPICS READINGS  
1 Linear equations 1.1-2.1 Solve a system of linear equations
n linear equations, n unknowns
Row Picture: one equation at a time
* Column Picture (rows and columns
Matrix Form
2 Solve by Elimination using matrices 2.2-2.3 Method of Solution by "Elimination" to elimate a variable
Is Matrix a good or bad Matrix?
Matrix Modifications
Ax = b
3 Rules for Matrix operations (add, mul)
and inverses
2.4-2.5 Matrix Multiplication (4 ways)
Matrix Inverse of A AB A^T
Gauss-Jordan or fin A^01
4 Inverse and Results of Inverses 2.6 Inverse AB, A^T
Product of eliminaton matrices
A=LU (no row exchanges)
Gaussian Elimination
5 Transposes and Permutations 2.7-3.1 PA = LU
Vector spaces (Beginning of Linear Spaces)
Vector subspaces
Permutations P: execute row exchanges
A=LU
6 Vector spaces (and Subspaces) 3.1 What is a Vector Space (add and multipy vestors)
Multiply a vector bu a constant
All linear combinations stay in vector spaces
Vector subspace inside a Vector space
7 The Nullspace: Solving Ax = 0 3.2 Computing the "Nullspace (Ax=0)" and "Column space:
Pivot variables & Free variables
Special Solutions - rref(A)=R
8 Rectangular PA = LU and Ax = b 3.3 (3.4) Find a Completely solution to linear equations, Ax=b
How many of solutions: 0, 1, 2, etc
Rank
Augmented matrix
Solvability Condition on b
Ax=b is solvable when bis in C(A)| i.e.
     If a combination of rows of A gives zero row
    Then the same combination of entries of b

 

9* Linear Independance 3.3 (3.4) A bunch of vectors being independent .. spanning a space
Basis and Dimension
a) More unknowns than rquations m < n with free variables

 

10 Basis and dimension 3.5 3.4
11 The four fundamental subspaces 3.6 3.5
12 Exam 1: Chapters 1 to 3.4    
13 Graphs and networks 8.2 3.5, 10.1
14 Orthogonality 4.1 4.1
15 Projections and subspaces 4.2 4.2
16 Least squares approximations 4.3 4.3
17 Gram-Schmidt and A = QR 4.4 4.4
18 Properties of determinants 5.1 5.1
19 Formulas for determinants 5.2 5.2
20 Applications of determinants 5.3 5.3
21 Eigenvalues and eigenvectors 6.1 6.1
22 Diagonalization 6.2 6.2
23 Markov matrices 8.3 10.3
24 Review for exam 2    
25 Exam 2: Chapters 1-5, 6.1-6.2, 8.2    
26 Differential equations 6.3 6.3
27 Symmetric matrices 6.4 6.4
28 Positive definite matrices 6.5 6.5
29 Matrices in engineering 8.1 10.2
30 Similar matrices 6.6 6.2
31 Singular value decomposition 6.7 7.1-7.2
32 Fourier series, FFT, complex matrices 8.5, 10.2-10.3 10.5, 9.2-9.3
33 Linear transformations 7.1-7.2 8.1-8.2
34 Choice of basis 7.3 8.3
35 Linear programming 8.4 10.4
36 Course review    
37 Exam 3: Chapters 1-8 (8.1, 2, 3, 5)    
38 Numerical linear algebra 9.1-9.3 11.1-11.3
39 Computational science See the Web site for 18.085  
40 Final exam