linear-algebra - Linear Independence &...

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Unformatted text preview: Linear Independence & Differential Equations Definition 1. A set of functions { y 1 ,...,y n } , y i : I R , is linearly independent iff you cannot write one function as a linear combination of the others. More precisely, iff c 1 y 1 + + c n y n = 0 = c 1 = c 2 = = c n = 0 . (1) Use this definition to prove the special cases for linear independence: (i) If z ( x ) 0 is the zero function, then any collection containing it is depen- dent. (ii) If there are only two functions, then they are linearly independent iff nei- ther is a constant multiple of the other. (iii) If a collection is dependent and you add something to it, the enlarged collection will also be dependent. (iv) If a collection is independent and you remove an element of it, the reduced set will also be independent. Let X = ( a,b ) be any open interval in R , including the possibility that X = R . Suppose that y 1 ,y 2 ,...,y n are solutions of an ODE and that each has ( n- 1) con- tinuous derivatives on X . Then the Wronskian is defined (for any x X ) to be the determinant W...
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linear-algebra - Linear Independence &...

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