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:$\begin{equation} \left[ \begin{array}{c} S_t(1) \\ S_t(2) \\ \vdots \\ S_t(\omega_N) \end{array} \right] = \begin{bmatrix} f(x_1, x_2, 1) & f(x_2, x_3, 1) & \cdots & f(x_i, x_{i+1}, 1) \\ f(x_1, x_2, 2) & f(x_2, x_3, 2) & \cdots & f(x_i, x_{i+1}, 2) \\ \vdots & \vdots & \ddots & \vdots \\ f(x_1, x_2, \omega_N) & f(x_2, x_3, \omega_N) & \cdots & f(x_i, x_{i+1}, \omega_N) \\ \end{bmatrix} \times \left[ \begin{array}{c} S_1 \\ S_2 \\ \vdots \\ S_i \end{array} \right] \label{equationsystem} \end{equation}$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

 

[blue_raver22]

Hello,

Based on the information provided, this system of equations does not qualify as linear. A system of linear equations is one where the variables appear only in a linear fashion, meaning they are not raised to any powers or involved in any other non-linear operations. In this case, the function f(x_i, x_{i+1}, \omega_N) contains exponential terms, which makes it non-linear.

To solve a system of linear equations, the inverse of the coefficient matrix can be used, as you mentioned. However, in this case, since the function matrix is not linear, the inverse cannot be calculated using traditional methods.

I would suggest exploring other methods for solving non-linear systems of equations, such as using numerical methods or approximation techniques.

I hope this helps clarify the concept of linear systems of equations. Let me know if you have any other questions.

Best,