1 bezy- 2 – 1 (Holder inequality) reen Recorder + 1. Show that || || , ||1|| - ||*||, are vector norms. 2. Show that if

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1 Bezy 2 1 Holder Inequality Reen Recorder 1 Show That 1 Are Vector Norms 2 Show That If 1 (86.54 KiB) Viewed 22 times

1 bezy- 2 – 1 (Holder inequality) reen Recorder + 1. Show that || || , ||1|| - ||*||, are vector norms. 2. Show that if x and y are two vectors, then ||||||- ||||||||x + y||* ||*|| + || y ||. 3. If x and y are two n -vectors, then prove that (a) ſx” y] < ||xl|, 12 1. | 1 p 9 (b) ||xy*, * ||*|| ||||(Schwarz inequality). 4. Let x and y be two orthogonal vectors. Then prove that x + y = 12+ 5. Prove that for any vector x, we have (a) ||«ll2 = || 411, <Vn ||2ll2 (b) - ||*|| = || | . = ||2|| 지 6. Prove that || A||- ||1||. , ||1||, are matrix norms. n 7. Prove that the vector length is preserved by orthogonal matrix multiplication. That is, if