ABSTRACT
We observed in (4.7) and (4.20) that not only every matrix A ∈ K m×n determines a linear mapping K n → K m via x ↦ Ax but that also every linear mapping ƒ: K n → K m arises in this way. Now it is of utmost importance to observe that matrices can be used not only to represent linear mappings between the spaces K n but that in fact all linear mappings between finite-dimensional vector spaces can be represented by matrices. To see why this is the case, we first set up a correspondence between the elements of an arbitrary n-dimensional vector space V over a field K with the elements of K n which generalizes the identification of equivalence classes of arrows with elements of ℝ3 in section 1.
