ABSTRACT
Many of the objects of interest in differential geometry on manifolds are expressed properly in the context of multilinear algebra. This chapter discusses linear algebraic concepts that are not commonly included in a first linear algebra course. In applications of bilinear forms to geometry, linear transformations that preserve the form play a key role. The chapter explains the dual of a vector space, the Hom-space, and bilinear forms on vector spaces. It explores those constructions through what is called the tensor product. Tensor fields on manifolds will play a key role in describing structures of interest on manifolds. The difference in how coordinates of covectors change versus how coordinates of vectors change under a basis change is of foundational importance with implications reaching into many areas of mathematics.
