
2. A vector space that is also a {ring}, where the vector space and the ring share the same addition operation and are related in certain other ways. An example algebra is the set of 2x2 {matrices} with real numbers as entries, with the usual operations of addition and matrix multiplication, and the usual scalar multiplication. Another example is the set of all polynomials with real coefficients, with the usual operations. In more detail, we have: (1) an underlying set, (2) a field of scalars, (3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a vector space, (4) an operation of addition of members of the underlying set, whose input is an {ordered pair} of such members and whose output is one such member, just as in a vector space or a ring, (5) an operation of multiplic スポンサード リンク
