lecture_25 (dragged) 1

# lecture_25 (dragged) 1 - 2 MA 36600 LECTURE NOTES MONDAY...

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2 MA 36600 LECTURE NOTES: MONDAY, MARCH 23 Express this as a product of matrices as follows: f 1 ( t ) f 2 ( t ) ··· f n ( t ) f (1) 1 ( t ) f (1) 2 ( t ) ··· f (1) n ( t ) . . . . . . . . . . . . f ( n 1) 1 ( t ) f ( n 1) 2 ( t ) ··· f ( n 1) n ( t ) k 1 k 2 . . . k n = 0 0 . . . 0 . n × n matrix on the left is nonsingular if and only if the only solution is k 1 = k 2 = ··· = k n = 0. Hence its determinant, namely W ° f 1 ,f 2 ,...,f n ± ( t ), is nonzero if and only if { f 1 ,f 2 ,...,f n } is a linearly independent set. This completes the proof. Homogeneous Equations with Constant Coeﬃcients. Consider now the constant coeﬃcient equation a 0 d n y dt n + a 1 d n 1 y dt n 1 + ··· + a n 1 dy dt + a n y =0 . For example, when n = 2, we made a guess that solutions to the equation a 0 y °° + a 1 y ° + a 2 y =0 are in the form y ( t )= e rt for some constant r . We will try this in general. Denote the linear operator L [ y ]= a 0 d
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