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Mathematics
Differential Equations
Fundamental solutions and fundamental matrices
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[QUOTE="faradayscat, post: 5445003, member: 579452"] I'll put you in context for the sake of simplicity before asking my question. Say we have the following homogeneous linear system: [B]x[/B]'=A[B]x[/B] Let A be 2x2 for simplicity. Then the general solution would look like: [B]x[/B](t) = α[B]a [/B]+ β[B]b [/B] And a fundamental matrix would be: Ψ(t) = ( [B]a[/B] , [B]b[/B] ) What confuses me is this: I tried making a new fundamental matrix by replacing the first column of Ψ(t) by a linear combination of the general solution, something like: [B]x[/B](t) = 2[B]a[/B] + 4[B]b [/B] Now my new fundamental matrix looked like this: Ψ(t) = ( 2[B]a[/B] + 4[B]b [/B], [B]b [/B]) And expanding the following expression: [B]x[/B](t)=Ψ(t)[B]c[/B], where [B]c[/B] is the vector of constants, I found out that I get the same general solution [B]x[/B](t), with different eigenvectors (however they were simply scalar multiples of the eigenvectors of the matrix A) My question is this, are linear combinations of the fundamental set of solutions also a fundamental set of solutions? Like, would { 2[B]a[/B] + 4[B]b [/B], [B]b [/B]} also be a fundamental set of solutions? I guess it would because they are linearly independent... If not, why do we call Ψ(t) a fundamental matrix when we can build one using linear combinations of the fundamental set of solutions? All these questions confuse me, I just need some clarification. Thanks in advance! [/QUOTE]
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Fundamental solutions and fundamental matrices
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