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Unformatted text preview: LS.5 Theory of Linear Systems 1. General linear ODE systems and independent solutions. We have studied the homogeneous system of ODEs with constant coecients, (1) x = A x , where A is an n n matrix of constants ( n = 2 , 3). We described how to calculate the eigenvalues and corresponding eigenvectors for the matrix A , and how to use them to find n independent solutions to the system (1). With this concrete experience solving loworder systems with constant coecients, what can be said in general when the coecients are not constant, but functions of the independent variable t ? We can still write the linear system in the matrix form (1), but now the matrix entries will be functions of t : x = a ( t ) x + b ( t ) y a ( t ) b ( t ) x x (2) y = c ( t ) x + d ( t ) y , y = c ( t ) d ( t ) y , or in more abridged notation, valid for n n linear homogeneous systems, (3) x = A ( t ) x . Note how the matrix becomes a function of t we call it a matrixvalued func tion of t , since to each value of t the function rule assigns a matrix: a ( t ) b ( t ) t A ( t ) = c ( t ) d ( t ) In the rest of this chapter we will often not write the variable t explicitly, but it is always understood that the matrix entries are functions of t . We will sometimes use n = 2 or 3 in the statements and examples in order to simplify the exposition, but the definitions, results, and the arguments which prove them are essentially the same for higher values of n . Definition 5.1 Solutions x 1 ( t ) , . . . , x n ( t ) to (3) are called linearly dependent if there are constants c i , not all of which are 0, such that (4) c 1 x 1 ( t ) + . . . + c n x n ( t ) = 0 , for all t. If there is no such relation, i.e., if (5) c 1 x 1 ( t ) + . . . + c n x n ( t ) = for all t all c i = , the solutions are called linearly independent , or simply independent ....
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This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Fall '09 term at MIT.
 Fall '09
 vogan
 Linear Systems

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