5. Let A € Mnxn(C) with characteristic polynomial p(x) = cxºII-1(d; – x) and li + 0, Vi, a E Z>o. Show that if dim(ker(A))+k=n, then A= C2 for some complex matrix C.
5.3 Let F be an ordered field, let d > 0, and suppose that d does not have a square root in F. Let F(Vd) denote the set of all a+bvd, with a, b e F, where vd is a square root in some extension field of F (a) Show that F(Va) is a field. (b) Show how to define an ordering on FVa), with vd> 0, such that it becomes an ordered field
Exercise 6. Show that if f(x) > 0 for all x e [a, b] and f is integrable, then Sfdx > 0.
8. Let f (x) e, 0 > 0; x> 0 (1 1 +e (a) Show that f(x) is a probability density function (b) Find P(X> x) (c) Find the failure rate function of X
Prove or Disprove: Let p E P(F) and suppose that deg p > 1 and p is irreducible. Then p(a)メ0 for all a E F.
6 Points Let F be the vector field represented in the figure: P(-1,1) 1907/1X X Q3.1 3 Points O2d-Curl F(0,0) > 0 O2d-Curl F(0,0) = 0 O2d-Curl F(0,0) < 0 Q3.2 3 Points OV.F(0,0) > 0 OV.F(0,0) = 0 OV.F(0,0) < 0
8. Consider the circuit: t=0 212 f(t)t 222 11 Pilt) 1F yt (a) Show that the transfer function of the circuit for t > 0 is † (s) = F(*) 452 +55+2 (b) What are the characteristic modes of the circuit (c) Determine the response y(t) for t > 0 if f(t) = 1, y(0-) = 1 V and i(0-) = 0.
Let Y be a random variable with probability density function, pdf, f(y) = 2e-2y, y > 0. Determine f (U), the pdf of U = VY.
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...