Mathematical+concepts

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

Info icon This preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
System as a mathematical mapping System H x(t) x[n] y(t) y[n]
Image of page 2
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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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 0 0 0 0 1 ; 0 0 0 0 0 1 0 0 0 0 0 1 n = = I AI A O O O O O
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 6
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 0 ), then output is y 3 (t)=y(t-t 0 )
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Example LTI or Not? ( 29 2 2 0 K r r r r r θ θ θ - = - + = & && && & &
Image of page 8
Why is this important? Most enzyme kinetic systems are not linear or even time-invariant So why?
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 10
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.
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 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
Image of page 12
11 12 1 1 1 21 22 2 2 2 1 2 1 0 0 0 1 0 0 1 1 n n m m mn m m a a a b d a a a b d m n a a a b d < L L L L M M O M
Image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 14
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern