Extra Credit 2
These are a few miscellaneous interesting things that I
could have discussed in chapter 1 but did not have time. I st
(1)(2 points) The first thing I want you to do is reexamine
the norm extra credit I assigned a couple weeks ago, which no one did in full
generality. So for the vector 
(never mind if it’s a bit different from the old vector) I
want you to
(a) show that 
for all
, and
(b) show that 
.
(2)(2 points) You know of course that there are nonzero 2x2
matrices that do not have inverses, and they don’t always commute, and crazy
stuff like that. However if you consider
the class of 2x2 matrices of the form 
, you’ll get a nice surprise.
First note that this matrix can be written as 
where 
is the identity and 
. Note that 
, which is like the 
definition of the
complex number i.
(a) Check
that if 
is any other matrix of this form, then 
; i.e. this family of 2x2 matrices commutes.
(b) Check that 
, just as for complex numbers 
. Therefore any
nonzero matrix in this class does have an inverse.
It turns out that all the rules of complex arithmetic hold
for matrices of this form. The discovery
of this matrix representation, in terms of real numbers only, of complex
numbers helped convince skeptics that complex numbers were indeed real.
(3) (2 points) After the discovery of complex numbers, many
mathematicians searched for systems of numbers with more than two
components. This search ultimately led
to general matrix algebra in the 1850’s and later, but in 1843William Rowan
Hamilton had a spectacular insight that a 4-component system was possible. He had spent much of the 1830’s trying to
find a 3-component system, hoping to find a natural way to do algebra with
three-dimensional space. But he, like
everyone else, was unsuccessful. With
our current knowledge of matrices and complex numbers, it turns out to be
easy. The only tricky part is that we’re
going to mess with the notation a bit.
We’re going to denote the identity matrix by U (for unit) instead of I,
and we’re going to define 
. I want you to check
(a) that 
, so that I, J, K can
all be thought of as square roots of -1,
(b) that 
, which you hopefully recognize as the vector cross-product
relations for the coordinate unit vectors 
. (So he did succeed in developing a useful 3d
algebra after all!) And
(c) that any nonzero quaternion,
a matrix of the form aU+bI+cJ+dK with
real numbers a, b, c, d (not all
zero) has an inverse.
(1)(3 points) This problem is based on my Matrix Optics handout under the Handouts link. I’ll be happy to discuss any questions you have in my office hours.
A Galilean telescope consists of two lenses (focal lengths f and g ) separated by a distance d . The light source is so far away that we can basically assume the rays arrive from the left as a parallel bundle with the same slope regardless of height. They are then refracted by the primary lens (focal length f ), travel distance d along the telescope axis, and are refracted again by the secondary lens. When the telescope is `focused’, the rays are again parallel as they exit the secondary. Mathematically that means that the slope of the exiting rays is independent of the height of the incoming rays at the primary.
(a) Calculate the matrix (in terms of f, g, d ) that converts the height and slope of a ray entering the primary lens into its height and slope as it leaves the secondary.
(b) What relationship between f, g, d ensures that the telescope is focused?
(c) What is the magnification (ratio of slope of entering ray to slope of exiting ray)? Observe that the ratio is negative (inverted image) when both lenses are convex (positive focal lengths), and positive when one is convex and the other concave.
Normal terrestrial telescopes are
arranged to give uninverted images because otherwise we would feel weird
looking at upside-down scenes. But ast