One of the important results in linear algebra is the rank nullity theorem. Here I am going to present a proof of it which is slightly less known. The reason I like this proof is because it ties together many concepts and results quite nicely, and also because I independently thought of it.
The theorem (as is well known) says that if are vector spaces with and a linear map then .
In this proof I will further assume that is finite dimensional with dimension . A more general proof can be found on wikipedia.
We start by fixing two bases of and and obtain a matrix of relative to these bases. (Each is a row matrix). Then our theorem basically translates to . We let and claim that .
Clearly if then and so so that each is orthogonal to . Hence . Conversely if then so that , i.e. following which .
Now it only remains to invoke the result for any subspace of an inner product space to conclude that . In other words .