Any step explanation required then comment below..
Understand every step carefully which is in inside note box...ok.
6. Let F : Rn → Rn, n 1, 2, 3, , be a conservative force. Suppose that a mass m is moving in Rn under the influence of F according to Newton's second law of motion, F ma. Show that the total mechanical energy of the mass, E(t), is constant in time over its trajectory x(t). (You will, of course, have to define Et),) Be sure that you denote all vector quantities as vectors, e-g, or z, and dot...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x, where r is real. For each n let yn = (1/n) E*j. Show that yn + x. HINT: (xj – a) Let e >0 and use the definition of convergence. Split the summation into two parts and show that each is < e for all sufficiently large n.
QUESTION 2. Suppose that S= {n e Z (Si Z)(n = 7i + 3)}, T= {n E ZI (3j e Z)(n = 7j - 4)}. Prove that S=T.
PART C ONLY! Thank you.
14. Fix a non-zero vector n R". Lot L : Rn → Rn be the linear mapping defined by L()-2 proj(T), fa TER or all (a) Show that if R", Such that oandj-n -0, then is an eigenvector of L What is its cigenvaluc? (b) Show that is an cigenvector of L. What is its cigenvalue? (c) What are the algebraic and geometric multiplicities of all cigenvalues of L?
14. Fix a non-zero vector n...
A square matrix E∈Mn×n(R) is idempotent if E2=E. It is symmetric if E = tE. (a) Let V⊆Rn be a subspace of Rn, and consider the orthogonal projection projV:Rn→Rn onto V. Show that the representing matrix E = [projV]EE of proj V relative to the standard basis E of Rn is both idempotent and symmetric. (b) Let E∈Mn×n(R) be a matrix that is both idempotent and symmetric. Show that there is a subspace V⊆Rn such that E= [projV]EE. [Hint: What...
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that {z.) is Cauchy. Is this result true under the condition Irn +1-Fml < rt Let xi = 1 and xn +1 = (Zn + 1)/3 for all n e N. Find the first five terms in this sequence. Use induction to show that rn > 1/2 for all n and find the limit N. Prove that this sequence is non-increasing, convergent,
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.
(a) Show that Rn \ {0} is path connected when n ≥ 2. (b) Show that Rn R when n ≥ 2.
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Exercise 6.5.28 Let S (z, y, z) e R3 1 z? + уг + z2-1,#2 0} be the upper hemisphere of the unit sphere in R3. For each of the following integrals, first predict what the integral will be, and then do the computation to verify your prediction 22. 222. 1U. JS Definition 6.5.9 Let S,T C(RT, R). The wedge product of S and T is the alternating bilinear form SAT : Rn × Rn → R given...