What is the significance of theoretical linear algebra in machine learning/computer vision research? I am a computer science research student working in application of Machine Learning to solve Computer Vision problems.
Since, lot of linear algebra(eigenvalues, SVD etc.) comes up when reading Machine Learning/Vision literature, I decided to take a linear algebra course this semester. 
Much to my surprise, the course didn't look at all like Gilbert Strang's Applied Linear algebra(on OCW) I had started taking earlier. The course textbook is Linear Algebra by Hoffman and Kunze. We started with concepts of Abstract algebra like groups, fields, rings, isomorphism, quotient groups etc. And then moved on to study "theoretical" linear algebra over finite fields, where we cover proofs for important theorms/lemmas in the following topics:

Vector spaces, linear span, linear independence, existence of basis.
  Linear transformations. Solutions of linear equations, row reduced
  echelon form, complete echelon form,rank. Minimal polynomial of a
  linear transformation. Jordan canonical form. Determinants.
  Characteristic polynomial, eigenvalues and eigenvectors. Inner product
  space. Gram Schmidt orthogonalization. Unitary and Hermitian
  transformations. Diagonalization of Hermitian transformations.

I wanted to understand if there is any significance/application of understanding these proofs in machine learning/computer vision research or should I be better off focusing on the applied Linear Algebra?

Personally, linear algebra will be very important for your research in machine learning. Because it provides not only the best interpretaion of many real problems, but also gives the easy solutions to these problems, such as linear regression model and linear classification model. In addition, with the knowledge of linear algebra, it will be easier to study other courses including statistics which is important for machine learning. 
Maybe, I can recommend a book for you on linear algebra, "Linear Algebra Done Right" edited by Sheldon Axler.
Best!
