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Unformatted text preview: Results from Linear Algebra We first state the basic existence/uniqueness theorem for 1st-order linear systems and examine some of the consequences. Theorem (Existence/Uniqueness) The 1st-order linear homogeneous IVP y = A y , y ( x ) = y always has a unique solution, which is defined for every x in the inverval (- , ) . Remark: In the above theorem, by y we mean the vector of initial values of the functions y 1 ,y 2 ,...,y n , that is to say the vector y = y 1 y 2 . . . y n . Recall the following definitions from linear algebra: Definition A vector space is a set of objects that is closed under the operations of addition and scalar multi- plication (and a distributive property). A set of vectors v 1 ,v 2 ,...,v n are linearly-independent if whenever c 1 v 1 + c 2 v 2 + + c n v n = 0 for some scalars c 1 ,c 2 ,...,c n , then necessarily all of the c i must be 0. The vectors v 1 ,v 2 ,...,v n form a basis for the vector space V if they are linearly-independent and every vector v in V can be written in the form v = c 1 v 1 + c 2 v 2 + +...
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