3.3we - 3.3: LINEAR INDEPENDENCE AND THE WRONSKIAN KIAM...

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3.3: LINEAR INDEPENDENCE AND THE WRONSKIAN KIAM HEONG KWA Generally, two functions f ( t ) and g ( t ) are said to be linearly indepen- dent on an interval I if the linear relation (1) c 1 f ( t ) + c 2 g ( t ) = 0 , where c 1 and c 2 are constant scalars, holds for all t I only if c 1 = c 2 = 0. Otherwise, f ( t ) and g ( t ) are said to be linearly dependent on I . In the latter case, there are constants c 1 and c 2 with the property that either c 1 6 = 0 or c 2 6 = 0 such that the linear relation (1) holds on I . Note that if c 1 6 = 0, then f ( t ) = - ± c 2 c 1 ² g ( t ), while if c 2 6 = 0, then g ( t ) = - ± c 1 c 2 ² f ( t ) throughout the interval I . In other words, f ( t ) and g ( t ) are linearly dependent on I if one of them is a constant multiple of the other throughout the interval I . A relationship between the Wronskian and the linear independence of two functions is provided by the following theorem. Theorem 1. 1 If f ( t ) and g ( t ) are differentiable functions on an open interval I such that W ( f,g )( t 0 ) 6 = 0 for some point t 0 in I , then f ( t ) and g ( t ) are linearly independent on I . Equivalently, if f ( t ) and g ( t ) are linearly dependent on I , then W ( f,g )( t ) = 0 for all t in I . Proof. Let c 1 and c 2 be constants such that c 1 f ( t ) + c 2 g ( t ) = 0 on I . Then, clearly, c 1 f 0 ( t ) + c 2 g 0 ( t ) = 0 on I as well. In particular, c 1 f ( t 0 ) + c 2 g ( t 0 ) = 0 , (2a) c 1 f 0 ( t 0 ) + c 2 g 0 ( t 0 ) = 0 . (2b) Multiplying (2a) and (2b) by g 0 ( t 0 ) and g ( t 0 ) respectively gives c 1 f ( t 0 ) g 0 ( t 0 ) + c 2 g ( t 0 ) g 0 ( t 0 ) = 0 , (3a) c 1 g ( t 0 ) f 0 ( t 0 ) + c 2 g ( t 0 ) g 0 ( t 0 ) = 0 . (3b) Date : January 17, 2011. 1 This is theorem 3.3.1 in the text. 1
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2 KIAM HEONG KWA Now substracting (3b) from (3a) yields c 1 W ( f,g )( t 0 ) = 0. Hence if W ( f,g )( t 0 ) 6 = 0, then c 1 = 0. Likewise, we can show that if W ( f,g )( t 0 ) 6 = 0, then c 2 = 0 as well. Hence if W ( f,g )( t 0 ) 6 = 0, then f ( t ) and g ( t ) are linearly independent. ± Remark 1. The converse of theorem 1 does not hold without additional conditions on the pair of functions involved. That is to say, it is not true in general that two differentiable functions f ( t ) and g ( t ) with a vanishing Wronskian on an open interval must be linearly dependent on the same interval. See the following example due to Giuseppe Peano, a famous Italian mathematician. Example 1. Consider the differentiable functions f ( t ) = t 2 | t | and g ( t ) = t 3 on R . It can be shown that f 0 ( t ) = ( - 3 t 2 if t < 0 , 3 t 2 if t 0 . Hence
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3.3we - 3.3: LINEAR INDEPENDENCE AND THE WRONSKIAN KIAM...

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