Transcription of LS.6 Solution Matrices - MIT Mathematics
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LS.6 Solution Matrices - MIT Mathematics
Solution MatricesIn the literature, solutions to linear systems often are expressed using square matricesrather than vectors. You need to get used to the before, we state thedefinitions and results for a 2 2 system, but they generalize immediately ton Fundamental return to the system(1)x =A(t)x,with the general Solution (2)x=c1x1(t) +c2x2(t),wherex1andx2are two independent solutions to (1), andc1andc2are arbitrary form the matrix whose columns are the solutionsx1andx2:(3)X(t) = (x1x2) =(x1x2y1y2).Since the solutions are linearly independent, we called them in afundamentalset ofsolutions, and therefore we call the matrix in (3) afundamental matrixfor the system (1).
where x1 and x2 are two independent solutions to (1), and c1 and c2 are arbitrary constants. We form the matrix whose columns are the solutions x1 and x2: (3) X(t) = (x1 x2) = x1 x2 y1 y2 . Since the solutions are linearly independent, we called them in LS.5 a fundamental set of
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