Is this a linear system of equations?

[nde]

Hello everyone!

I have a question on whether a system of equations can be classified as linear. I have the following matrix:<br /> \begin{equation}<br /> \left[ \begin{array}{c} S_t(1) \\ S_t(2) \\ \vdots \\ S_t(\omega_N) \end{array} \right] = <br /> \begin{bmatrix} f(x_1, x_2, 1) &amp; f(x_2, x_3, 1) &amp; \cdots &amp; f(x_i, x_{i+1}, 1) \\ <br /> f(x_1, x_2, 2) &amp; f(x_2, x_3, 2) &amp; \cdots &amp; f(x_i, x_{i+1}, 2) \\<br /> \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ <br /> f(x_1, x_2, \omega_N) &amp; f(x_2, x_3, \omega_N) &amp; \cdots &amp; f(x_i, x_{i+1}, \omega_N) \\<br /> \end{bmatrix} <br /> \times <br /> \left[ \begin{array}{c} S_1 \\ S_2 \\ \vdots \\ S_i \end{array} \right]<br /> \label{equationsystem}<br /> \end{equation}<br />where f(x_i, x_{i+1}, \omega_N) is a non-linear function containing two exponential terms and S_i is unknown. Does this system of equations qualify as linear if I know x_i, x_{i+1} and \omega_N and plug it into f(x_i, x_{i+1}, \omega_N) to yield a numerical value (real number)?

If this is true, I should be able to figure out S_i by taking the inverse of the function marix and multiplying both sides with it.

I greatly appreciate your input. Thank you in advance for taking the time to answer this.

Kind regards.

 

[jedishrfu]

Yes if the f(...) are considered constant coefficients then you can do the matrix inversion but if they are somehow dependent on the S unknowns then all bets are off.

 

[nde]

Thanks for your reply. What do you mean when you say that the S_i are somehow dependent on the f? Could you please illustrate it with a simple example?

 

[jedishrfu]

say Si is a function xi and as is the f(...) functions then its no longer a linear system.

Wikipedia has a writeup on linear systems that outlines the definition:

http://en.wikipedia.org/wiki/Linear_system_of_equations