B1;2c 18.06 Linear Algebra

18.06 Linear Algebra

Essays and Papers

  • Goals of the Linear Algebra Course (html)
  • The Language of Linear Algebra (pdf)
  • A Factorization Review (ps, pdf)
  • Glossary for Linear Algebra (ps, pdf)
  • Linear Algebra in a Nutshell (ps, pdf)
  • Too Much Calculus (PDF)
  • A 100 dollar bet (pdf)
  • Starting with Two Matrices (pdf)
  • The Four Fundamental Subspaces: 4 Lines (pdf)
  • Fourier Sine Series Examples (pdf)
  • Notes on function spaces, Hermitian operators, and Fourier series (pdf)
  • A summary of how the properties of different matrices are reflected in the eigenvalues/eigenvectors: (ps, pdf).
  • Pascal Matrices (article by Alan Edelman and Gilbert Strang): (ps, pdf)
  • Linear Algebra and Music: (pdf)   This fascinating article, with MATLAB codes for music and for telephone tones and for recovering answering machine information, was contributed by Derrick Smith of Laney College in Oakland. Thank you!!
  • A Basis for 3 by 3 Symmetric Matrices (ps, pdf)
  • Gram-Schmidt in 9 Lines of MATLAB(ps, pdf)
  • Gram-Schmidt orthogonalization -- a nice example(ps, pdf)
  • "Operations Required in Matrix Elimination" by Kundan Chintamaneni (pdf)
  • Eigenvectors and Eigenvalues website

  • Extras

  • The SVD at work(ps, pdf): These are the pictures resulting from the best rank 1, rank 5, rank 10, rank 20 and rank 50 approximations to a 499 by 750 black-and-white intensity matrix. The approximations were obtained by keeping the k largest singular values in the SVD. The bottom right picture is the original one.

  • (Question from Professor Ian Christie, West Virginia University) Find unit vectors h(t) and m(t) in the direction of the hour and minute hands of a clock, where t denotes the elapsed time in hours. If t = 0represents noon then m(0) = h(0) = (0,1). At what time will the hands of the clock first be perpendicular? At what time after noon will the hands first forma straight line? In the dot product m(t) * h(t),remember that sin x sin y + cos x cos y = cos(x - y).   Solution: (ps, pdf)

  • Multiplication by Columns! The multiplication Ax produces a combination
    of the columns of A. If the vectors a1, a2, ... , an are those columns, then

               Ax = x1a1 + ... + xnan = combination of columns (in the column space!)

  • Demos

  • Matrix Multiplication, A = LU, and PA = LU Demo: Interactive demo
  • Eigenvalue Demos

             2x2 Eigenvectors   This 3-minute demo shows eigenvectors of 2 by 2 matrices
    Watch the whole thing, or by individual parts:   Part1  Part 2  Part 3  Part 4  Part 5  Part 6  Part 7

             The Power Method   Powers AnV lead toward the top eigenvalue/eigenvector

  • Eigenvalue Mini-Lectures
  •          Full Lecture (all eight together)

        Or to view individually (about 2 minutes each)


             Eigenvectors and Trace



             Differential Equations


             Positive Definite


  • JAVA DEMOS (these are interactive, without voice explanation)
  • The Java® Demos were developed by Pavel Grinfeld.

  • Other Demos