Exercise Set 2

Exercise Set 2 - Exercise Set 2 Math 4027 Due 1 Determine...

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Unformatted text preview: Exercise Set 2 Math 4027 Due: February 10, 2005 1. Determine, with justification, whether each of the following lists of functions is linearly dependent or linearly independent. (a) ϕ 1 ( x ) = e x , ϕ 2 ( x ) = e x +2 I Solution. These functions are linearly dependent since if c 1 = e 2 and c 2 =- 1, then c 1 ϕ 1 ( x ) + c 2 ϕ 2 ( x ) = e 2 ϕ 1 ( x ) + (- 1) ϕ 2 ( x ) = e 2 e x- e x +2 = 0 , for all x . J (b) ϕ 1 ( x ) = e x , ϕ 2 ( x ) = e 2 x I Solution. These functions are linearly independent since if c 1 e x + c 2 e 2 x = 0 for all x , then dividing by e x gives c 2 e x =- c 1 . Differentiate to get c 2 e x = 0, and divide by e x to get c 2 = 0. Since c 2 = 0 it follows c 1 e x = 0, and dividing by e x gives c 1 = 0. Hence, c 1 e x + c 2 e 2 x = 0 for all x = ⇒ c 1 = c 2 = 0 , as required. Alternatively, note that W ( ϕ 1 , ϕ 2 )(0) = fl fl fl fl 1 1 1 2 fl fl fl fl = 1 6 = 0 . J (c) ϕ 1 ( x ) = sin x , ϕ 2 ( x ) = sin( x + 2) I Solution. These functions are linearly independent since W ( ϕ 1 , ϕ 2 )(0) = fl fl fl fl 0 sin2 1 cos2 fl fl fl fl =- sin2 6 = 0 ....
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Exercise Set 2 - Exercise Set 2 Math 4027 Due 1 Determine...

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