HW6 - AMS210.01. Homework 6 Andrei Antonenko Due at the...

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AMS210.01. Homework 6 Andrei Antonenko Due at the beginning of the class, April 28, 2003 In this homework we will consider the following scalar products In R n : h ( x 1 ,x 2 ,...,x n ) , ( y 1 ,y 2 ,...,y n ) i = x 1 y 1 + x 2 y 2 + ··· + x n y n ; In M m,n : h A,B i = tr( AB > ); In C [ a,b ]: h f,g i = R b a f ( t ) g ( t ) dt . In this homework there are a lot of extra-credit problems of different levels of difficulty. You should understand perfectly how to solve standard problems, and proceed to extra-credit problems only after that! The problems from exam and quizes will include only standard problems. 1. Compute h 3 u - 5 v, 2 u + v i if h u,u i = 5 , h u,v i = 1 and h v,v i = 2. 2. Compute the following scalar products: (a) h (2 , 1 , - 3 , 1) , (0 , 4 , 2 , 2) i in R 4 with standard scalar product. (b) h 2 t + 1 ,t 2 i in the space C [0 , 1]. (c) h 2 t + 1 ,t 2 i in the space C [ - 1 , 1]. 3. Find norms and normalizations of the following vectors: (a) (4 , 2 , 2) in R 3 with standard scalar product. (b) t 2 in C [0 , 1]. (c) t 2 in C [ - 1 , 1]. 4. Find the cosines of the angles between the following vectors: (a) (0 , 2) and (3 , - 3). (b) t 2 and t 2 + 1 in C [0 , 1]. (c) ˆ 1 2 3 4 5 6 ! and ˆ 4 5 6 1 2 3 ! in M 2 , 3 . 5. Find all constants a such that (a) vectors ( a, 2) and ( a, - 8) are orthogonal in R 2 . 1
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(b) vectors t 2 and t 2 + a are orthogonal in C [0 , 1]. 6. Determine the distances between the following points:
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HW6 - AMS210.01. Homework 6 Andrei Antonenko Due at the...

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