I'll be studying Brannan's Geometry and Lang's Introduction to Linear Algebra for one university course.
I would like to know if you can you suggest some books that offer a unique perspective on the material covered by such books and that can be therefore used as effective companions.
This unique perspective may include, for example, the emphasis of the geometric side of linear algebra
For a different take on linear algebra, try Practical Linear Algebra, A Geometry Toolbox by Farin and Hansford. (reviews)
For geometry, try Geometry: Euclid and Beyond by Hartshorne. (MAA review)