By Giuliano Sorani

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Indeed the zero vector is a linear combination of any set of vectors, with all scalars 0. 7. The arrow from 0 lo C represenls 4u + 2v. 8. Johannes Brahms! 9. Seven years 10. Joseph R. "Joey• Smallwood, the Premier who brought Newfoundland into Confederation with canada In 1949. 11. Cioacchino Rossini. ) True/False Questions Decide, with as little calculation as possible, whether each of the following statements is true or false and, if you say "false,• explain your answer. ) 1. IfA = (1,2) andB = (-3,S), the vector Ai=[-~].

BB] Suppose vectors u and v are pictured by arrows as shown. Ftnd a formula for a vector whose arrow bisects angle AOB. -·• I ? ,,. --· 0 . )( B v 44. Let u and v be two vectors that are not parallel and let w = u + v. Explain why w bisects the angle between u and v. 45. et u and v be nonzero vectors and set w = llvll u + II uII v. Show that w bisects the angle between u and v. D 46. (a) Prove that the sum of the lengths of the diago· nals of a rhombus is at least as great as the sum of the lengths of two opposite sides.

6The geometrlc mean appears tn the work of Euc:l1d. /fib. Chapter l. 26 Theorem (The Triangle Inequality). II u + v II andv iD R". :S II uII + llvll for any vectors u Proof. The proof follows directly from the fact that llxll 2 = x. x for any vector x. > (llull + llvll) 2 , so Lhe resull follows by laking square roots. • At step (3), we had nu+ vu 2 = 11u11 2 + 2u • v + 11v11 2• Since vectors u and v are orthogonal if and only if u · v = 0, we obtain the well-known Theorem of Pythagoras (in any Euclidean space Rfl).

### An introduction to real and complex manifolds. by Giuliano Sorani

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