linear-algebra

# linear-algebra - Linear Independence& Differential...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

linear-algebra - Linear Independence& Differential...

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

View Full Document
Ask a homework question - tutors are online