Let k be an infinite field (a) Show that any straight line in k” is irreducible...
Let F be a field and V a vector space over F with the basis {v1, v2, ..., vn}. (a) Consider the set S = {T : V -> F | T is a linear transformation}. Define the operations: (T1 + T2)(v) := T1(v) + T2(v), (aT1)(v) = a(T1(v)) for any v in V, a in F. Prove tat S with these operations is a vector space over F. (b) In S, we have elements fi : V -> F...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
Assume b.1 is proven. Please help prove b.2
(b) Let f: V V be any linear map of vector spaces over a field K. Recall that, for any polynomial p(X) = 0 ¢X€ K[X] and any vE v p(X) p(u) 2ef°(v). i-0 The kernel of p(X) is defined to be {v € V : p(X) - v = 0}. Ker(p(X)) (b.1) Show that Ker(p(X)) is a linear subspace of V. When p(X) = X - A where E K, explain...
Please solve the exercise 3.20 .
Thank you for your help !
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Review. Let M be a o-algebra on a set X and u be a measure on M. Furthermore, let PL(X, M) be the set of all nonnegative M-measurable functions. For f E PL(X, M), the lower unsigned Lebesgue integral is defined by f du sup dμ. O<<f geSL+(X,M) Here, SL+(X, M) stands the set of all step functions with nonnegative co- efficients. Especially, if f e Sl+(X,...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...