Can linear algebra used to deal with non linear systems?

[dexterdev]

Hi all,
Can linear algebra used to deal with non linear systems? and why linear algebra is 'linear'? :(

-Devanand T

 

[krome]

"Linear" in "Linear Algebra" means "closed under addition". In physics a "linear system" is one that satisfies the superposition principle, which is just the physics way of saying closed under addition. This means that if $S_1$ and $S_2$ are two possible states of the system (i.e. two possible solutions to the equations of motion), then $S_1 + S_2$ is also a possible state of the system (i.e. it also solves the equations of motion). The same word, "linear", is also used to describe the equations of motion in this case; one says that the equations of motion are linear if their solutions satisfy this superposition principle. This use of the word simply generalizes the fact that if you add two points on a line, you end up with a point on the same line.

Linear techniques (e.g. Fourier transformation, perturbation theory, etc.) can be used to approximate the behavior of non-linear systems over sufficiently brief time periods, but most non-linear systems can only be "solved" numerically and display complicated chaotic behavior.

 

[lurflurf]

linear algebra is linear because it deals with functions such that

f(a*x+b*y)=a*f(x)+b*f(y)

Many nonlinear systems have linear or approximately linear parts so linear algebra is still useful.

 

[dexterdev]

Thankyou guys...you people cleared my doubt

 

[epenguin]

A good deal of biomath is linearising about stationary points and considering their stability, which helps give a qualitative picture of the overall behaviour.