### 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.

### 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.