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:
    • nullrangeplot.m
    • vector3d.m
• 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
    • Scripts for today: random_solvable.m, random_unsolvable.m, gausselim.m
    • Output of Gaussian elimination
    • Gauss-Jordan elimination
    • Echelon form / Reduced echelon form
    • Unsolvable systems
    • Pivot variables and free variables
  • 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
    • QR factorization
    • Script written in lecture: QRplayground.m

  • 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
    • Code: newtonpolyval.m, forwardsubstitute.m
    • Polynomial interpolation
    • Interpolation in Newton form
    • Problems with interpolation

  • Lecture #22 --- August 15, 2008
    • Approximating derivatives with quadratic interpolation
    • Error in approximations