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!!
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
