Hint It is a fact that if a 1 and a 2 are non negative real numbers then a 1 a

# Hint it is a fact that if a 1 and a 2 are non

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Hint: It is a fact that if a 1 and a 2 are non-negative real numbers, then ( a 1 + a 2 ) α a α 1 + a α 2 for α 1 . (c) Optional bonus question : Prove the fact in the hint above. 4. One way to visualize a norm in R 2 is by its unit ball , the set of all vectors such that k x k ≤ 1. For example, we have seen that the unit balls for the 1 , ‘ 2 , and norms look like: Given an appropriate subset of the plane, B R 2 , it might be possible to define a corre- sponding norm using k x k B = minimum value r 0 such that x rB, (1) where rB is just a scaling of the set B : x B r · x rB. 1 Last updated 12:11, August 29, 2019

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-1 -1 1 1 -1 -1 1 1 -1 -1 1 1 B 1 = { x : k x k 1 1 } B 2 = { x : k x k 2 1 } B = { x : k x k 1 } (a) Let x = 3 2 . For p = 1 , 2 , , find r = k x k p , and sketch x and rB p (use different axes for each of the three values of p ). (b) Consider the 5 shapes below. (1,3) (1,-3) (-1,3) (-1,-3) (1,0) (-1,0) (0,-1) (0,1) B a B b (1,-1) (-1,1) (-1,-1) (1,1) (-1,0) (0,-2) (1,0) (0,2) (2,-2) (1,1) (-1,-1) (-2,2) B c B d B e Determine the j for which k·k B j is a valid norm. In the cases where k·k B j is not a valid norm, explain why. The most convincing way to do this is to find vectors for which one of the three properties of a valid norm are violated. 5. Let A be the 2 × 2 matrix A = 1 2 4 4 - 1 / 2 1 / 2 . For x R 2 , define k x k A = k Ax k 2 . (a) Show that k · k A is indeed a valid norm. (b) Sketch the unit ball B A = { x : k x k A 1 } corresponding to k · k A . Feel free to use MATLAB. 2 Last updated 12:11, August 29, 2019
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