What's the precise/exact definition of an inconsistent system of linear equations (including the source)
I noticed that the definitions in linear algebra, in general, need to be very precise as the omission of a single detail can have a significant effect on my overall understanding of certain concepts, so I'd also appreciate it if there you can include a source/sources that contain all the definitions of, for example, linear algebra.
There is no such thing as an "inconsistent system of equations". Each equation $\Phi_i(x)=0$, whether linear or not, defines a certain subset $A_i$ of the environment under consideration, and a system of $n$ such equations then captures the intersection $\bigcap_{i=1}^n A_i$ of these sets. It may happen that this intersection is empty; no big deal.
An example: Each of the three equations $$x-2y+5z=7,\quad 5x-8y+15z=1,\quad -2x+3y-5z=2$$ defines an plane $A_i\subset {\mathbb R}^3$ in an innocent way, but the intersection of these three planes is empty. There is nothing "inconsistent" here, Wikipedia notwithstanding.