b) Provide example in each defined term in (a).6. a) State the Fundamental Theorem of Galois theory.b) Outline the proof of (a).7. Answer the following:a) Show that the splitting field of f(x)=³-2 over Q is Q(2,w), where w = e.b) Find [Q(2,w): Q] and a basis for Q(2,w) over Q.c) Determine the Galois group of Q(2,w) over Q. To which group it is isomorphic to?d) Determine the subgroups of the Galois group in (c) that fixes Q(V2), Q(w), Q(√2w), andrespectively.e) Draw the lattice diagram of the Galois group and the corresponding tower of fields.1 Add fileubmit

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Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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discrete math
Let x=(1,1,1,1)^T and y=(8,2,2,0)^T (a) Determine the angle theta between x and y (b) find the vector projection p of x onto y (c) verify that x-p is orthognal to p (d) compute ||x-p||, ||p||, ||x|| and verify that the pythagorean law is satisfied
(a) Let f: V > V be any linear map where V is a vector space of dimension n over a field K. Suppose there is a vector v E V so that v, f(v),... ,f"-1 (v) are independent, hence form a basis of V. (a.1 Show that there exist a, ai, ..., an-1 E K so that f"(v)= an-1f"(v) + ... + af(v) + a0v. (a.2) Use (a.1) to find the matrix of f under the basis {v,f(v),... ,f"-1 (v)} (a.3) Find the characteristic polynomial of f
Q#3: Let V= P(R), and for PO) EV, define f EV by f,(PW)= [ PE)dt and to (Pos)= Pt)dt. Find a basis for V for which ct is the dual basis.
11. Let the complex field C be a vector space over the real field R and the conjugation operator T on C defined by T(z) = z. 1 a. Find the matrix of T relative to the usual basis {1, i}. b. Find the matrix of T relative to the basis {1+i,1+2i}.
' Define an inner product on the vector space P2(t) as following 3 (p(), g(t) -Σ α.)g (a,) i=0 where ao, a1, a2, a3 = -3, –1, 1,3 Construct the orthogonal bases from p(t) = 1, q(t) = t,r(t) = t? space P2(t).
Functional analysis
1. Let V be the vector space over R, consisting of all complex polynomials of degree at most 2; i.e. V = {ao + a₁x + a₂x² | a₁ € C}. (a) Find a basis for V. Give proper reasons. (b) Find the dimension of V.
are not vector spaces? 23) Which of the following subsets of C[-1,1] 1 (x)dx 0 f(x) E CI-1,1]: a) -1 {f(x)e C-1,1]: f(-1) = f(l)} {f (x) E C[-1,1]: f(x) 0 for xe[-1/2,1 / 2]} {f(x) e Cl-1,1]: f (1) = 1} e) {f(x)e CI-1,1]: f(l) = 0} b) c) d)
Section 2.1: Exercices:17
Which of the following statements is (are) true if B is a basis for the vector space V (a) In some situations, a plane in R6 can be isomorphic to R4 (b) If x is in V and B contains n vectors, the the B- coordinate vector of x is in R (c) The vector spaces P5 and RS are isomorphic. (d) If PB is the change of coordinates matrix, then [x]B = PB x for x in V (e) A coordinate mapping is a bijective mapping O All of the above Noneoftheabove Options a, b, c and e Options a, b and e Options b, c and e Options a, b, d and e Options b, c, d and e Options a, and b

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6. a) State the Fundamental Theorem of Galois theory. b) Outline the proof of (a).