Mathematical+concepts

Mathematical+concepts - L 7: Linear Systems and Metabolic...

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Unformatted text preview: L 7: Linear Systems and Metabolic Networks Reading: Chapter 3.1,3.2 Integration of Ordinary Differential Equations dy’/dt = f(t,y) Marco Lattuada System as a mathematical mapping System H x(t) x[n] y(t) y[n] Linear Equations • Form • System 1 1 2 2 n n a x a x a x b + + + = K 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 n n n n m m mn n m a x a x a x b a x a x a x b a x a x a x b + + + = + + + = + + + = K K M K Linear Systems 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 n n n n m m mn n m a x a x a x b a x a x a x b a x a x a x b + + + = + + + = + + + = K K M K = Ax b 11 12 1 1 21 22 2 2 1 2 n n m m mn m a a a b a a a b a a a b = = A b L L M M O M M L Matrices in review • Notation • Identity Matrix [ ] 11 12 1 1 21 22 2 2 1 2 n n ik m m mn m a a a b a a a b a a a a b = = = A b L L M M O M M L 1 1 ; 1 1 n = = I AI A O O O O O Matrix operations • Transpose: interchange all the rows and columns • Sum and difference [ ] [ ] 11 21 1 12 22 2 1 2 n T n T ik ki m m nm a a a a a a a a a a a = = = A L L M M O M L [ ] [ ] / / ik ik a b + - = + - A B Linear Time-invariant Systems • Linear: weighted sum of signals lead to output – Superposition holds – If input is x 3 (t)=ax 1 (t)+bx 2 (t), then output is y 3 (t)=ay 1 (t) +by 2 (t) • Time-Invariant: – If you put in the same input at any time, then you get the same output. – If input is x 3 (t)=x(t-t ), then output is y 3 (t)=y(t-t ) Example • LTI or Not? ( 29 2 2 K r r r r r θ θ θ- = - + = & && && & & Why is this important? • Most enzyme kinetic systems are not linear or even time-invariant • So why? Row Echelon Form • A matrix is in row-echelon form if – All zero rows are at the bottom of the matrix – If two successive rows are non-zero, the first one starts with more zeros than the one before it • Row-equivalent: if matrix A can be obtained from matrix B using basic row operations – Implies that A and B have the same solution sets. Vocabulary • If b =0, then system is homogeneous • If a solution (values of x that satisfy equation) exists, then system is consistent, else it is inconsistent. Solve linear system using Gaussian Elimination • Form Augmented Matrix, • Row equivalence, can scale rows and add and subtract multiples to transform matrix 11 12 1 1 1 21 22 2 2 2 1 2 1 1 1 2 2 2 1 1 1 1 1 1 n n m m mn m n n n n a a a b d a a a b d m n a a a b d d x d d x d d x d → = = = → = → = x L L L L M M O M M M M M O M L L L L M M M M O M L 11...
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This note was uploaded on 03/08/2010 for the course BCB 570 taught by Professor Juliedickerson during the Spring '10 term at Iowa State.

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Mathematical+concepts - L 7: Linear Systems and Metabolic...

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