### Homework assigned 11/4, due 11/13

There is a midterm next week on Tuesday the 11th. It will cover
finite-difference differentiation, integration, and linear algebra up
to and including today's lecture.

Please come see me if your notes from class are insufficient.
### Problem 1

Pick a 2 x 2 matrix, A. Plot the unit-ball in L^2 and plot the
image of this ball after A has been applied to it. In the L^2
norm, what is the furthest point from the origin, i.e. what is
the largest value of ||A x||_2 for ||x||_2 = 1? What is the
point nearest to the origin, i.e. what is the smallest value of
||A x||_2 for ||x||_2 = 1?
### Problem 2

For the 2 x 2 matrix A of problem 1, plot the unit-ball in L^1 and plot the
image of this ball after A has been applied to it. In the L^1
norm, what is the furthest point from the origin, i.e. what is
the largest value of ||A x||_1 for ||x||_1 = 1? What is the
point nearest to the origin, i.e. what is the smallest value of
||A x||_1 for ||x||_1 = 1?
### Problem 3

For the 2 x 2 matrix A of problem 1, plot the unit-ball in L^inf and
plot the image of this ball after A has been applied to it. In the
L^inf norm, what is the furthest point from the origin, i.e. what is
the largest value of ||A x||_inf for ||x||_1 = inf? What is the point
nearest to the origin, i.e. what is the smallest value of ||A x||_inf
for ||x||_inf = 1?
### Problem 4

Repeat Problem 1 for three other matrices. (Try to pick some
interesting ones.)
### Problem 5

Repeat Problem 1 for a 3 x 3 matrix A. Present the unit sphere in
L^2, and present its image using "sphere.m". Again, find the points
furthest from and nearest to the origin.
### Problem 6

A line in R^2 is a collection of points such that for some fixed a, b,
and c, the points on the line satisfy a x + b y = c. Prove that if
you start with a line then its image under a linear transformation A
is also a line.
### Problem 7

A circle in R^2 centered around the origin is a collection of points
such that for some fixed r, the points on the circle satisfy x^2 + y^2
= r^2. Prove that if you start with a circle then its image under a
linear transformation A is an ellipse (the set of points that
satisfies a x^2 + b y^2 = 1 for some positive a and b).
### Problem 8

A plane in R^3 is a collection of points such that for some fixed a,
b, c and d, the points in the plane satisfy a x + b y + c z = d.
Prove that if you start with a plane then its image under a linear
transformation A is also a plane.