Math410 (Bayly) Homework 2 (due Friday 19 July)

If you’re not used to basic matrix multiplication, refresh your memory by reading section 1.2 and doing the various parts of problem 1.2.7. (In the unlikely event that you’re still uncertain about what a transpose is, you can look at problems 1.6.1 and 1.6.2 also.)

The following problems to turn in are just a few of the many educational problems in the sections we’ve covered. I encourage you to try problems besides the ones I assigned, and definitely come in to office hours and ask me about anything weird you encounter.

To turn in, do 1.8.4, even though it’s messy, and probably not exactly how the author intended, and 1.8.8(c, i) on page 69. You’ll have to consider some separate cases in 1.8.4, and it will be worth more points. Also 1.3.22 and 1.3.25 (page 20), 1.5.4 (page 35) and 1.5.24e (page 41), and 2.21c (page 123).

And finally, 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 is a new matrix you’ve never seen before. However, you can check (don’t turn in) that , which is kind of like the definition of the complex number i.

What I would like you to check (and turn in!) is that , just as for complex numbers . Therefore any nonzero matrix in this class does have an inverse, and 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.

EXTRA CREDIT

(1)(5 points) 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.

(d) Normal terrestrial telescopes are arranged to give uninverted images because otherwise we would feel weird looking at upside-down scenes. But astronomical Galilean telescopes were usually made with two convex lenses and gave inverted images. Why do you think this choice was commonly made?

(2)(5 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 first to check that , so that I, J, K can all be thought of as square roots of -1. I’d also like you to check 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!) Last, I’d like you to confirm that any nonzero quaternion, a matrix of the form aU+bI+cJ+dK with real numbers a, b, c, d, has an inverse.