MATH 475 - Spring 2022 – Homework 3
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MATH 475 - Spring 2022 – Homework 3
Questions
1. (a) Show that the ‘0-norm’ is not a norm. Which of the three properties of a norm does it satisfy?
(b) For 0 < q < 1, define the ‘q-norm’ by
|z|q = ╱ inⅠ← Ixi Iq \ ←/q .
Is this a norm? Explain your answer.
(c) Draw a picture of the unit ball of the ‘q-norm’ for several different q .
2. For 0 < q < 1, prove that the qth power of the ‘q-norm’ satisfies the triangle inequality |z + y|q(q) s |z|q(q) + |y|q(q) , z, y e =n .
3. This question concerns a generalization of Theorem 4 of Set 6 of the lecture notes. Let A e =k 之n and 0 < q s 1. Show that every s-sparse vector z e =n is the unique solution of minimization problem
|5 |q subject to A5 = Az,
if and only if
|uS |q < |uS∈ |q , Vu e N(A)\{0},
and all subsets S S {1, . . . , n} with ISI s s.
Hint: use question 2.
2022-03-14