Using linear algebra to tell when your derivation is impossible?

[phasor]

Sorry if this is the wrong place to ask this, but I think linear algebra is the best place to ask my question. Feel free to move this thread elsewhere if I am wrong.

I would like to know how I can use linear algebra to help me figure out when I am deriving an equation if the derivation I would like to have is possible. So, if I have a relation with variables x and y, is it possible to express y in terms of only x? The other way around?

Keplers Equation makes a good example: M = E - e*Sin(E). (e is eccentricity). M can be expressed in terms of E, but because this equation is transcendental E cannot be expressed in terms of M. That's easy enough to see here, but what if I have complicated expressions for M and E, and I am trying to derive Kepler's equation? Is there a theorem I can apply to figure out, before doing lots of algebra, that E cannot be expressed in terms of only M?

Thanks for the help.

 

[SteamKing]

If your equation has lots of transcendental functions in it, that's a good indicator that no closed form solution is possible.

To answer your basic question, no, linear algebra cannot be used to determine if a closed-form derivation can be developed for a particular variable.

 

[phasor]

Maybe Kepler's Equation was a bad choice for an example. But there are plenty of equations loaded with transcendental functions that have closed form solutions. There's no method of discovering if a problem has a closed form solution other than algebra + inspection?