linear algebra also studies vector spaces additional structure inner product inner product example bilinear form gives vector space geometric structure allowing definition length angles formally ''inner product'' map math langle cdot cdot rangle v times v f math satisfies following three axioms vectors '''u''' '''v''' '''w''' ''v'' scalars ''a'' ''f'' ref name jain ref ref name prugovec ki ref conjugate symmetry math langle mathbf u mathbf v rangle overline langle mathbf v mathbf u rangle math '''r'''
Formal science Mathematics Pure mathematics Algebra Linear algebra