Im aware of the question Derivation of IV estimator? on this site.
Im interested however in obtaining the way we derive it using linear algebra.
$$\beta^{IV}=(Z'X)^{-1}Z'Y$$
the reason why I ask is to obtain a clearer picture of how exactly we get this equation.
One possible answer to this question is that we know that OLS is the solution to $$ (y-X\beta)'(y-X\beta) $$ w.r.t. to $\beta$.
The 2SLS estimator $\beta^{2SLS}=(X'P_ZX)^{-1}X'P_Zy$ is the solution to $$ (y-X\beta)'P_Z(y-X\beta), $$ where $P_Z$ is the (symmetric) projection matrix $P_Z=Z(Z'Z)^{-1}Z'$.
The IV formula you quote results under exact identification, so that $X'Z$ is square and the expression for 2SLS simplifies accordingly, noting that $$ (X'Z(Z'Z)^{-1}Z'X)^{-1}=(Z'X)^{-1}(Z'Z)(X'Z)^{-1}. $$