hw6 - Math 307: Assignment 6 Please look at the Course News...

Math 307: Assignment 6
Please look at the Course News to see which of the follwing problems need to be solved. Definitely,
solving all of the problems would be helpful for the midterm exam but it is not mandatory.
1. Compute the 1,2 and infinity norms for the following vectors. Is it always the same norm
that is biggest? (The 2-norm is the standard Euclidean norm)
2. Find four vectors in two dimensions whose 1, 2 and infinity norms the same.
3. Suppose you were doing a problem each entry v
i
in a vector [ v
1
,v
2
,...,v
n
] is positive and
represents the yearly production of one of n factories. Which of the three norms we
introduced have natural interpretations in this context.
4. Draw a picture of the “unit circle” for the 1,2, and infinty norms in two dimensions. By
“unit circle” we mean the set of all vectors whose norm is equal to one.
5. Recall that the Euclidean distance between two vectors v and w is bardbl v − w bardbl where we use
the standard norm. If we use the 1 -norm or ∞ -norm in this formula, we obtain different
distance functions. Consider vectors whose entries are either 0 or 1 (like [0 , 1 , 1 , 0 , 1] ).
Describe in words the meaning of the 1 -distance and the ∞ -distance between two such
vectors.
6. Use MATLAB/Octave to compute both sides of the triangle inequality bardbl x + y bardbl ≤ bardbl x bardbl + bardbl y bardbl
for some random vectors x and y and for each of the norms ( 1 , 2 and ∞ ) that we defined.
Verify that for these examples the inequality is true.
7. The p -norm of a vector v = [ v
1
,v
2
,...,v
n
] T for 1 ≤ p ≤ ∞ is defined to be
bardbl v bardbl
p
=
parenleftBigg
n summationdisplay
i =1
| v
i
|
p