18.06 Linear Algebra, Spring 2012

SYLLABUS: Current (Spring 2012) class schedule and syllabus (PDF file)

Essays and Papers

Goals of the Linear Algebra Course (html)

A Factorization Review (ps, pdf)

Glossary for Linear Algebra (ps, pdf)

Linear Algebra in a Nutshell (ps, pdf)

Too Much Calculus (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)

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, ... , a_n are those columns, then

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