m25old - Math 2025, Quiz #2 Name: 1) Find the average value...

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Unformatted text preview: Math 2025, Quiz #2 Name: 1) Find the average value of the numbers 1 , 3 ,- 1 , 2. Answere: 2) Find the Haar wavelet transform of the initial data- s 2 = (3 , 1 , 2 , 6). Answere: 3) Assume that the Haar wavelet transform of the initial data- s 2 = ( a (2) ,a (2) 1 ,a (2) 2 ,a (2) 3 ) produces the result- s = ( 3 , ,- 2 , 1). Find the initial array- s 2 = ( a (2) ,a (2) 1 ,a (2) 2 ,a (2) 3 ). Answere: 2 Math 2025, Quiz #3 Name: 1) Calculate the in-place fast Haar wavelet transform of the sample- s = (5 , 1 , 2 , 4). Answer:- s (0) = 2) Assume that the in-place fast Haar wavelet transform of a sample- s = ( s ,s 1 ,s 2 ,s 3 ) produces the result (4 , 1 , 2 , 2). Apply the inverse transform to reconstruct the sample- s Answer:- s = 2) Evaluate the following: (1) (1 + 2 i )(2- i ) = (2) Write the complex number 1 1+4 i in the form x + iy . 1 1+4 i = (3) (5 + 3 i ) = (4) What is Re(2- 3 i ) = and Im(2- 3 i ) = (5) What is | 3 + 4 i | = 3 Math 2025, Quiz #4 Name: The inner product on R n is <- x ,- y > = x 1 y 1 + ... + x n y n , on C n is <- z ,- w > = z 1 w 1 + ... + z n w n where stands for complex conmjugation. The inner product on the space of continuous functions C ([ a,b ] , C ) is given by < f,g > = R b a f ( t ) g ( t ) dt . 1) Evaluate the integral R 1 (1- it ) 2 dt = 2) Evaluate the inner products: (1) < (2 , 1 ,- 1) , (3 ,- , 1) > = (2) < (1 + i, 1- i, ( i, 2 ,- 1 + i ) > = 3) Evaluate the norm of the vector (1 ,- 2 , 1). The norm is: 4) Let a = 0 and b = 1. Evaluate the the following inner products: (1) < t, 1 + it > = (2) < t,e t > = 4 Math 2025, Quiz #5 Name: 1 ) Which of the following sets of vectors is linearly independent? (1) (1 , , 1) , (0 , 1 , 0) , (1 , ,- 1). They are paar wise orthogonal and hence linearly indepen- dent. (2) (1 , , 0) , (1 ,- 1 , 1) , (1 , 2 , 2) , (1 , 2 , 5). Four vectors in R 3 are always l inearly dependent . Another solution is to notice that (1 , 2 , 5) =- 9 4 (1 , , 0) + 3 2 (1 ,- 1 , 1) + 7 4 (1 , 2 , 4) 2) Which of the following sets is a basis for R 2 ? (1) (1 , 0) , (1 ,- 1). Basis . The two vectors are perpenticular and hence linearly independent. In R 2 any set of two linearly independent vectors is a basis. (2) (1 , 1) , (1 , 0) , (1 ,- 1). Not a basis . The set is generating but not linearly independent. 3) Which of the following sets is a basis for R 3 ? (1) (1 , , 0) , (1 , 1 , 0) , (1 , 1 , 1). The set is linearly independent. The equation c 1 (1 , , 0) + c 2 (1 , 1 , 0) + c 3 (1 , 1 , 1) = ( c 1 + c 2 + c 3 ,c 2 + c 3 ,c 3 ) = (0 , , 0) give: c 1 + c 2 + c 3 = 0 c 2 + c 3 = 0 c 3 = 0 . The last equation gives c 3 = 0. Then the second implies that c 2 = 0, and then finally the first gives c 1 = 0. Hence all the constants have to be zero....
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m25old - Math 2025, Quiz #2 Name: 1) Find the average value...

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