Unformatted text preview: c + c 1 x + c 2 x 2 + Â·Â·Â· + c m âˆ’ 1 x m âˆ’ 1 , vanishes on this interval (the exponential factor, e rx , is never zero). But, as we have proved in Problem 27, Section 6.1, the system of monomials { 1 , x, . . . , x n } is linearly independent on any interval. Thus, the linear combination (6.3) vanishes on an inteval if and only if it has all zero coeï¬ƒcients, i.e., c = c 1 = . . . = c m âˆ’ 1 = 0. Therefore, the system { e rx , xe rx , . . . , x m âˆ’ 1 e rx } is linearly independent on any interval, in particular, on ( âˆ’âˆž , âˆž ). 355...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at Berkeley.
 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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