Open mapping theorem,which is taught in functional analysis,states that if $X$ and $Y$ are two Banach spaces and $T:X\to Y$ be a surjective bounded linear operator ,then the map $T$ is open.Now I think this theorem has beautiful applications in linear algebra:
Suppose $S=\{(x+z,z,y+z): x^2+y^2+x^2<1/4\}\subset \mathbb R^3$,then this set is open.This can be shown directly but we can also use open mapping theorem.We should try to define linear map in such a way that $(x,y,z)\mapsto (x+z,z,y+z)$,this can be done if we define $T(x,y,z)=(x+z,z,y+z)$,notice that $T:\mathbb R^3\to \mathbb R^3$ defined in this way is surjective.Now note that $T(A)=S$ where $A=\{(x,y,z):x^2+y^2+z^2<1/4\}=B(0,\frac{1}{2})$ is open.So,by open mapping theorem we can conclude that $S$ is open.
Is it not a nice application of open mapping theorem.I am asking this because I just observed this but there could be other simple ways to deal with this problem.