chapter3.page16

chapter3.page16 - on [a, b], one need to show that the only...

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16 1. BRIEF INTRODUCTION TO VECTORS AND MATRICES Here are some sets of linearly independent functions that we en- counter in solving a system of differential equations, assume k 1 ,k 2 , ··· ,k n are different numbers, - { t k 1 ,t k 2 , ··· ,t k n } . - { e k 1 t ,e k 1 t , · ,e k 1 t } . - { sin( k 1 t ) , sin( k 2 t ) , · , sin( k n t ) } . - { cos( k 1 t ) , cos( k 2 t ) , · , cos( k n t ) } . - The mixing of above sets. - For each above set, when multiplying each element by an com- mon nonzero factor, we get another linearly independent set. The following screen shot displays a heuristic Mathcad function that tries to determine if a given set of functions are linearly independent. Figure 5. Calculus tool bar One warning, the result of the program is not very reliable, the user should check the result manually to confirm the result. To manually check if an set of functions are linearly independent
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Unformatted text preview: on [a, b], one need to show that the only solutions are C 1 = 0 ,C 2 = , ,C n = 0 . if equation (1) holds for all t in [a, b], which requires strong algebraic skill. One method is to choose n dierent numbers { t 1 , t 2 , , t n } from [a, b] and using the functions to create an matrix, the compute the determinant of the matrix A = ( f i ( t j )) , if the determinant is not zero, the functions are linearly independent, but if the determinant is zero, it is inconclusive(most likely are linearly dependent). Example 3.2 . Determine if f 1 ( t ) = t 2-2 t + 3 , f 2 ( t ) = 2 t 2-5 t-6 , and f 3 ( t ) = 5 t 2-11 t + 4 are linearly independent....
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