NAME STUDENT NO: DEPARTMENT Q-2) Let F be a field and h Fr], deg h >...
Q-) Let F be an object ond V is a finite dimensional vector Space on the object. . that if v is linear trons formation, ronkt is zero a) Show or 1. b) If Liv> v is linear tronsformation, show that ker L c ker L² and L(v) 2 L² (v). ( Note : L²=LoL and ker L, be defined as subspace of L.).
Let F be a finite field with q elements. a) S -1 for every a*0 in F. how that a-1 either f* 0 or deg(f*)<q, and f* induces the same function on F as f does. function on F, then f=g. b) Let j(X)E F[X]. Show that there exists a polynomial /*(X)EF[X] such that c) Show that if two polynomials f and g, each of degree <g, induce the same
Let F be a finite field with q elements. a)...
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...
2. Let R be an integral domain containing a field K as a unital subring. (a) Prove that R is a K-vector space (using addition and multiplication in R). (b) Let a be a nonzero element of R. Show that the map is an injective K-linear transformation and is an isomorphism if and only if is invertible as an element of R. (c) Suppose that R is finite dimensional as a K-vector space. Prove that R is a field.
Show step by step please, I need A, B and C, THANKS!
Let (V,<,>) be a finite-dimensional Euclidean space n and let T be linear operator in V. Are the following statements true? Show your answer. (show by and =) A. T is orthogonal if and only if t preserves angles, that is, if e is the angle between a and B, then 0 is the angle between T(a) and T(B). B. T is orthogonal if and only if T...
Thee part question. Please answer all parts!
Let E be a field of characteristic p > 0 (we proved p must always be prime). Verify that the ring homomorphism X : Z → E determined by sending χ : 1-1 E (the unity in E) ( so x(n)-n 1E wheren1E 1E 1E (n-times), x(-n)- nle for any n 1,2,3,... and X(0) 0E by definition of χ) is in fact a ring homomorphism with ker(X) = pZ. Úse the fundamental homomorphism...
(2.2) Let a be a real number with 1<a< 2. Put f(x) = Q +r 1+2 (a) Show that f maps (1, 0) into (1, 0). (b) Show that f is a contraction on [1, ) and find its fixed point.
Let F be a field of characteristic p > 0. Show that f = t4 +1 € F[t] is not irreducible. Let K be a splitting field of f over F. Determine which finite field F must contain so that K = F.
P (0, +a) +40 +Q Calculate the magnitude of the force F acting on +Q, and on +4Q. a. b. Draw the force vector F on +Q, and on +4Q Find the electric Field vector E at (x, y)-(0, 0) and draw it. c. d. If the third charge Qo is placed at (x, y)-(0, 0), what is the force vector F acting on Qo? (Magnitude and direction) e. Find the electric Field vector E at point P (0, +a)....