To determine Whether the given functions are linearly dependent or linearly independent, if it is linearly dependent find the linear relation among them. Explanation Definition used: Linear dependence and independence: “The functions f 1 , f 2 , … , f n are said to be linearly dependent on an interval I if there exists a set of constants k 1 , k 2 , … , k n not all zero, such that k 1 f 1 ( t ) + k 2 f 2 ( t ) + ⋯ + k n f n ( t ) = 0 for all t in I . The functions f 1 , f 2 , … , f n are said to be linearly independent on I if they are not linearly dependent there.” Calculation: Consider the function f 1 ( t ) = 2 t − 3 , f 2 ( t ) = t 2 − 3 and f 3 ( t ) = 2 t 2 − t . Form the linear combination as follows. k 1 f 1 ( t ) + k 2 f 2 ( t ) + k 3 f 3 ( t ) = k 1 ( 2 t − 3 ) + k 2 ( t 2 + 1 ) + k 3 ( 2 t 2 − t ) = 2 t k 1 − 3 k 1 + t 2 k 2 + k 2 + 2 t 2 k 3 − t k 3 = ( − 3 k 1 + k 2 ) + ( 2 k 1 − k 3 ) t + ( k 2 + 2 k 3 ) t 2 Obtain the value of k 1 , k 2 , k 3 by setting all coefficients equal to zero

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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Chapter 4.1, Problem 7P

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Whether the given functions are linearly dependent or linearly independent, if it is linearly dependent find the linear relation among them.

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