The Rank-Nullity Theorem in Linear Algebra can be stated as follows:
Let ${\displaystyle T:V\to W} $ be a linear transformation between two vector spaces where ${\displaystyle T}$'s domain ${\displaystyle V}$ is finite dimensional. Then
$$ \mbox{Rank} (T)~+~\mbox{Nullity} (T)~=~\dim V. $$
For an invited talk that I shall be delivering soon, I like to give some engineering applications of Rank-Nullity Theorem, and I can't think of anything quickly.
One simple application is the homogeneous system $$ A x = 0 \tag{$\star$} $$ where $A$ is of size $m \times n$. Then Rank-Nullity Theorem can be applied for the linear map $T(x) = A x$ to deduce that the system ($\star$) admits only the trivial solution $x = 0$ if and only if $A$ has full column rank.
Similarly, using Rank-Nullity Theorem, we can easily establish the following:
The linear system $$ A x = b, \ \ \ (A \in R^{m \times n}, \ \ b \in R^n) $$ has a unique solution for every $ b \in R^n$ if and only if $A$ has full column rank.
I look forward to some more interesting applications, which are at a comprehensive level for the college students.