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LinearAlgebraApps

LinearAlgebraApps - Results from Linear Algebra We rst...

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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 0 ) = y 0 always has a unique solution, which is defined for every x in the inverval ( -∞ , ) . Remark: In the above theorem, by y 0 we mean the vector of initial values of the functions y 1 , y 2 , . . . , y n , that is to say the vector y 0 = y 0 1 y 0 2 . . . y 0 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 + · · · + c n v n for some scalars c 1 , c 2 , . . . , c n .

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LinearAlgebraApps - Results from Linear Algebra We rst...

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