Math 307: Assignment 6Please 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 normthat is biggest? (The 2-norm is the standard Euclidean norm)
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2.Find four vectors in two dimensions whose 1, 2 and infinity norms the same.3.Suppose you were doing a problem each entryviin a vector[v1,v2,...,vn]is positive andrepresents the yearly production of one ofnfactories.Which of the three norms weintroduced 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 isbardblv−wbardblwhere we usethe standard norm. If we use the1-norm or∞-norm in this formula, we obtain differentdistance functions.Consider vectors whose entries are either0or1(like[0,1,1,0,1]).Describe in words the meaning of the1-distance and the∞-distance between two suchvectors.6.Use MATLAB/Octave to compute both sides of the triangle inequalitybardblx+ybardbl ≤ bardblxbardbl+bardblybardblfor some random vectors x and y and for each of the norms (1,2and∞) that we defined.Verify that for these examples the inequality is true.7.Thep-norm of a vector v= [v1,v2,...,vn]Tfor1≤p≤ ∞is defined to bebardblvbardblp=parenleftBiggnsummationdisplayi=1|vi|p