What would you say links Differential Geometry & Linear Algebra? What would you say links differential geometry and linear algebra? how and why?
 A: As Daniel Schepler said the tangent space to a manifold at a point is a real vector space. The differential of a smooth map between manifolds is a linear map between the tangent spaces.
An important idea is "linearization"- that is you can learn a lot on the behaviour of a mapping through its differential (e.g the inverse function theorem).
It is worth noting that while linear algebra is extremely important for differential geometry, some of its stuff are less useful than others.
The theory of diagonalization of linear operators, which forms a large part of the material studied in linear algebra courses is less relevant in general (since it is on operators $V \to V$ while the differential of a map is a linear map between two different vector spaces).
(However, when studying fixed points of diffeomorphisms, the more "familiar" setting of maps $V \to V$ returns).
Most of the relevant linear algebra (besides the truly elementary ones), is related to alternating maps, exterior algebra and symmetric tensors etc, which are usually not studied in first year algebra courses.
