Math410 (Bayly) Homework 1 (due Monday 26 January)

The object of this set is to get you back up to speed on what matrices and vectors are, and the rules of their algebra. (I forgot to mention the operations of addition/subtraction and scalar multiplication, but these are simply defined elementwise, that is you do the familiar numerical operations on corresponding elements of the matrix or vector.)

I expect you to be able to do all the problems listed, but I’m going to collect only the ones in parentheses. The others are just for your own practice.

Section 1.4: 1, (2), 3, (4), 6, 8, (14), (19), 22, (23)

On problem 2 I’d also like you to calculate the max norms of each matrix and each vector (see handout on norms under the HANDOUTS link), and check that the max norm of the product is less than or equal to the product of the max norms of the factors.

Section 1.6: (13), 14, (15), 16

On problem 13 I’d also like you to calculate ATA, AAT, BTB, and BBT (imagine the T’s as superscripts, please). THEN calculate max norms of A, AT, B, BT, and all the products you calculated, and again verify that the multiplicative triangle inequalities hold. It’s interesting to see that ||AT|| is not necessarily equal to ||A||!

Extra Credit/ Honors Credit:

(1)(a) Show that the max norm for vectors satisfies the three norm axioms.

(b) Show that the explicit formula for the matrix max norm satisfies the matrix-vector multiplicative triangle inequality.

(c) How would you find a vector that makes the inequality an equality?

(2) 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?