Most important Linear Algebra theorems? I was reading up on symmetric matrices and the textbook noted that the following is a remarkable theorem:

A matrix $A$ is orthogonally diagonalizable iff $A$ is a symmetric matrix.

This is because it is impossible to tell when a matrix is diagonalizable, or so it seems. 
I haven't gotten to realize yet how important this is, but I will soon. What, in your opinion , is the most important linear algebra theorem and why?

Well since linear algebra is basically about studying linear functions in linear spaces, i would say the rank-nullity theorem. It basically limits the size of range in terms of the nullity, and vice-versa. THe nullity of a tranformation and the rank add up to the dimension of the space. I dont know if we can say which is the most important but this is a result worth understanding very well and it also has similar form in other structures.
