What is the significance of theoretical linear algebra in machine learning/computer vision research?

If you want to do advanced computer vision, and not just implement algorithms, you will need to understand advanced algebraic concepts for linear transformations. You will also need to understand a bit of measure theory and analysis.

Why?

Because research level computer vision involves the development of algorithms. The development of these algorithms necessarily invokes the structural properties of the mathematical objects; properties such as measure, convergence, isometry, isomorphism, etc.

Furthermore, say you have the mechanical skills to develop a computational method. Any true research-level effort is also expected to demonstrate a proof of convergence, establish a domain in which the method is efficacious, compare the method to prior methods, and fundamentally compare the weaknesses and benefits.

This requires at least a solid understanding of graduate-level analysis and 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!