lecture_16 - MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 23...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Page1 / 4

lecture_16 - MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 23...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online