V is an algebra of linear transformations, where in this algebra the product AB of two linear transformations A, B E 'c(V) is defined to be their composition; that is, AB is the linear transformation on V that acts on each vector v E V according to the rule (AB)v = A(Bv).
Both of these are quite special properties which are not shared by all linear transformations. As with the decomposition of a complex number into its real and imaginary parts, one can do the same for linear transformations. If T E SB(5)), then let ~T=~(T+T*) (the real part of T) , SST = (the imaginary part of T) , ;i (T - T*) so that Observe that ~ T and SS T are hermitian, where by a hermitian linear transformation is meant any A E SB(5)) for which A* = A. A hermitian matrix, in particular, is simply a matrix that is equal to its conjugate transpose.
2. If a polynomial f E IF [x], is irreducible, then the quotient ring IF [xli (f) is a field extension of IF. Recall that the Euclidean division algorithm in IF [xl asserts that if f is a fixed polynomial , say n f( x) = L aj x j , j=O then for every g E IF [x] there exist unique polynomials q (th e quotient) and r (the remainder) such that (i) g(x) = f( x)q(x) + r( x) , and (ii) r(x) = 0 or degr < degf , where deg f denot es the degree of the polynomial f. Assume that f is irreducible. , are equivalent modulo the ideal (f )) if and only if the polynomial f divides hI - h2.