AMATH 352
Summer Quarter, 2008
Applied Linear Algebra and Numerical Analysis
Lecture notes and MATLAB files
- Lecture #1 --- June 23, 2008
-
Course overview (obselete -- see the new syllabus on the main page)
- Linear algebra at a glance
- CVA package for MATLAB vector art --- CVA.zip
- Teaser for linear transformations
- Lecture #2 --- June 25, 2008
- Standard basis for R2
- Rotation as a linear transformation
- Matrix representation for a rotation
- Scaling in the horizontal and vertical directions
- Demo with CVA:
- A cat plotted with CVA
--- happycat.mat
- Demo of how rotation and scaling don't commute, using the cat
--- catdemo.m
- Lecture #3 --- June 27, 2008
- Inverses
- Scaling in perpendicular directions
- Scaling in non-perpendicular directions
- Lecture #4 --- June 30, 2008
- Inner products
- Length
- Orthogonal projections
- Invertibility
- Determinants of 2x2 matrices
- Finding scaling factors for a general transformation
- Lecture #5 --- July 2, 2008
- Inner product and length in R3
- Scaling vectors in R3 with linear transformations
- Lecture #6 --- July 7, 2008
- Linear systems of equations with examples
- Solving linear systems with Gaussian elimination
- Lecture #7 --- July 9, 2008
- Linear systems as matrix equations
- Subspaces
- Span
- Linear (in)dependence
- Lecture #8 --- July 11, 2008
- Basis
- Dimension
- Nullspace / nullity
- Range / rank
- Lecture #9 --- July 14, 2008
- Finding a basis for Nullspace(T)
- Finding a basis for Range(T)
- Rank-Nullity Theorem
- Geometry of the null space and range. These scripts are
explained in the lecture notes:
- Lecture #10 --- July 16, 2008
- Solving Tx=b when T has a nontrivial null space
- Orthogonalization, with pretty pictures on page 7 and 8
- Note: I talked about Gramm-Schmidt orthonormalization in this lecture, but I'm only putting it in the Lecture #11 notes. I think it really needed the longer intro to orthonormal sets of vectors.
- Lecture #11 --- July 18, 2008
- Orthonormal sets
- Why bother with orthonormalization?
- Gramm-Schmidt orthonormalization
- Lecture #12 --- July 21, 2008
- Programing Gaussian elimination in Matlab
- Matlab functions: scale.m,
swap.m, pivot.m,
gausselim.m (updated in Lecture #13)
- Lecture #13 --- July 25, 2008
- Lecture #14 --- July 28, 2008
- Code for today:
lecture14code.zip
- Constructing a particular solution to Ax=b
- Correspondence between pivot variables and Range(A)
- Constructing a basis for Nullspace(A) with free variables
- Lecture #15 --- July 30, 2008
- Square systems of equations
- Invertibility
- Finding the inverse of a matrix with Gauss-Jordan elimination
- Lecture #16 --- August 1, 2008
- Products of matrices
- LU decompositions
- Lower/upper triangular matrices
- Forward substitution / back substitution
- Lecture #17 --- August 4, 2008
- randomLU.m
- LU factorization in Matlab
- LUP factorization
- Lecture #18 --- August 6, 2008
- Lecture #19 --- August 8, 2008
- Code for today: backsubstitute.m,
noisyline.m
- Least squares problems: Approximately solving overdetermined systems
using the QR factorization
- Multiplication by Q' when Q has orthonormal columns
- Line fitting
- Lecture #20 --- August 11, 2008
- General polynomial fitting
- Functions as vectors (and vice versa)
- Lecture #21 --- August 13, 2008
- Lecture #22 --- August 15, 2008
- Approximating derivatives with quadratic interpolation
- Error in approximations