...F, then X- 1 is in A, so that Xs = s(X- 1), which lies in S. THEOREM 29. Let G be a torsion group. Let V be a finite -dimensional vector space over a field F. If F has characteristic p, assume also that G has no elements of order p. Then any representation of...

...viewed as column vectors, then Endz A, which consists of 2 x 2 matrices over Z, is noncommutative. 3. Let V be a finite dimensional vector space over a field F and T: F— » F, t; -» t;T; veV, a linear transformation. Then V is a right module over the ring...

...represent the data, use the fact that a flat h"l(y) is fully determined by y and the matrix of h.] 2. Let V be a finite dimensional vector space over a field F. Show that ACV is an affine flat if and only if every linear combination of vectors v,. e A with CXQ...

...defined from the inside by a point locus, and from the outside by a line envelope. 2. Dual spaces. Let V be a finite dimensional vector space over a field F. The dual of V is the vector space of linear maps of V into F. The space V and its dual, V , are isomorphic....

...prime characteristic p and * an element of order p in GL(V). Assume n > p. Prove dim(CV(*))> 1. 9. Let V be a finite dimensional vector space over a field F, / a nontrivial sesquilinear form on V with respect to an automorphism в of finite order m, and G =...

...this identifies the monomial e^ej^ • • • e*ik with (e^ei, • • • eik)* . Definition 7.3.6. Let V be a finite dimensional vector space over a field F of characteristic zero. A function / : V — > ¥ is a homogeneous polynomial of degree k on V if, relative...

...y', (cf. Lemma 3.2 below). VII, §2. THE CASIMIR DIFFERENTIAL OPERATOR We start with some multilinear algebra. Let V be a finite dimensional vector space over a field of characteristic ^ 2 (because we are going to use symmetric bilinear forms and quadratic forms and...

...pseudo-reflection. As we shall see in §14-3, this is only a true generalization for the "modular case". Let V be a finite dimensional vector space over a field F. Definition: A pseudo-reflection is a linear isomorphism s: V — > V that is not the identity map,...

...Show that the simple Z-modules are just the cyclic groups of the form Z(p) for pa prime. Problem 6.2.2 Let V be a finite dimensional vector space over a field F and let R = Lp(V) be the ring of linear transformations from V to V. Prove that V is a simple R-module....

....xm), v' = (y\ :y2, . . . ,у„) and £(v, v") = x\yi +X2y2 + ••• +xryr. Definition 15.5.7 : Let V be a finite dimensional vector space over a field F (of characteristic =/2). A quadratic form 9 : V — > F is such that 0(— v) = 0(v) for all v&V and 2£(и,...