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# lecture_16 - MA 36600 LECTURE NOTES MONDAY FEBRUARY 23...

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MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 23 Linear Independence Linear Independence. Let f = f ( t ) and g = g ( t ) be two differentiable functions. We say that f and g are linearly independent if the only constants c 1 and c 2 such that c 1 f ( t ) + c 2 g ( t ) = 0 for all t are c 1 = c 2 = 0. Otherwise, we say that f and g are linearly dependent . Note that linearly dependent means the functions are scalar multiples of each other i.e., g = λ f for some constant λ = - c 1 /c 2 . We can compute the Wronskian of f and g as a function of t : W ( f, g ) ( t ) = f ( t ) g 0 ( t ) - f 0 ( t ) g ( t ) . We will show that the following statements are equivalent: i. W ( f, g ) ( t 0 ) 6 = 0 for some t 0 . ii. W ( f, g ) ( t ) 6 = 0 for all t . iii. f and g are linearly independent. The phrase “the following are equivalent” has a precise mathematical meaning. The statement “if p then q ” can be thought of in terms of sets: this means p is “contained” in q , or “the validity of p implies the validity of q .” However note the conditional “if”: proposition p might not be true at all! The idea is if p is true, then so is q . The statement “ p if and only if q ” is called a biconditional because it consists of two statements: “if p then q ” and “if q then p .” When thinking of these as sets, this statement means logically p and q are equivalent. The phrase “the following are equivalent” means propositions p 1 , p 2 , . . . , p n are logically equivalent. That is, we have the statements “if p i then p j ” for all i and j . When n = 2, we have a biconditional. When n = 3 we have three propositions to consider, which means we have six conditionals: p 1 p 2 K ; p 3 s f Consider the following propositions: p 1 = “ W ( f, g ) ( t 0 ) 6 = 0 for some t 0 p 2 = “ W ( f, g ) ( t ) 6 = 0 for all t p 3 = “ f and g are linearly independent” We have already seen that p 1 = p 2 ; this is Abel’s Theorem. We show that p 2 = p 3 , so assume that W ( f, g ) ( t ) 6

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