Homework assigned 10/30, due 11/6

Problem 1

In class, I proved that
||x||_inf = max{ |x_1|, |x_2| }
is a norm.

Do one of the following:
1) Generalize my proof to n dimensions.
2) Prove that
||x|| = max{ 10|x_1|, |x_2| }
is a norm.

Problem 2

In class, I defined
||x||_1 = |x_1| + |x_2|
Prove || . ||_1 is a norm.

Problem 3

In class, we plotted balls of radius r around the origin using the ||.||_inf norm, the ||.||_1 norm, and the ||.||_2 norm. I defined the ||.||_p norm for 0 < p.

In the "Linear Transformations" link, I provide three *.m routines, L2_ball.m, L1_ball, and Linf_ball.m These routines take a radius and return a ball of that radius. Using matlab, write a subroutine L4_ball.m and use it to give me graphs of the balls of radii 1/2, 1, and 2 using the ||.||_4 norm.

The Sergel plaza in Stockholm has the shape of the unit ball in the 4 norm. And the Danish poet Piet Hein popularized this "superellipse" as a pleasing shape for objects such as conference tables.

Problem 4

Repeat the above exercise using the ||.||_1/4 norm.