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Unformatted text preview: 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 = f for some constant =- 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 ( t )- f ( t ) g ( t ) . We will show that the following statements are equivalent: i. W ( f,g ) ( t ) 6 = 0 for some t . 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 ) 6 = 0 for some t 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...
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This note was uploaded on 05/24/2011 for the course MA 36600 taught by Professor Ma during the Spring '09 term at Purdue University-West Lafayette.
- Spring '09
- Linear Independence