Is this a linear system of equations?

  • Thread starter nde
  • Start date
  • #1
nde
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Hello everyone!

I have a question on whether a system of equations can be classified as linear. I have the following matrix:


[itex]
\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}
[/itex]


where [itex] f(x_i, x_{i+1}, \omega_N) [/itex] is a non-linear function containing two exponential terms and [itex]S_i[/itex] is unknown. Does this system of equations qualify as linear if I know [itex]x_i, x_{i+1}[/itex] and [itex]\omega_N[/itex] and plug it into [itex] f(x_i, x_{i+1}, \omega_N) [/itex] to yield a numerical value (real number)?

If this is true, I should be able to figure out [itex]S_i[/itex] 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.
 
Last edited:

Answers and Replies

  • #2
12,241
5,953
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.
 
  • #3
nde
2
0
Thanks for your reply. What do you mean when you say that the [itex]S_i[/itex] are somehow dependent on the [itex]f[/itex]? Could you please illustrate it with a simple example?
 

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