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Section 71, Problem 9(a) The Frobenius norm (which is not a natural norm) is deflned for an n£n matrix A by jjAjjF = µ i=1 j=1 jaijj2 ¶1 2 Show that jj¢jjF is a matrix norm Solution For all n£n matrices A and B and all real numbers fi, we have (i) jjAjjF = µ i=1 j=1 jaijj2 ¶1 2.

Eqan eu e mb. Jan 27,  · Davneet Singh is a graduate from Indian Institute of Technology, Kanpur He has been teaching from the past 10 years He provides courses for Maths and Science at Teachoo. ) with (V;F s )Note that this is an atlas because F s is a homeomor phism from Bn = (V) to itself It is a smooth atlas because every. SOLUTIONS TO PROBLEMS FROM ASSIGNMENT 2 Problems 132d and 133d Statement Find general solutions of yu xy 2u x= xusing ODE techniques, as well as its particular solution satisfying the side conditions u(x;1) = 0 and u(0;y) = 0.

REAL ANALYSIS I HOMEWORK 3 2 then we have x= X n2N b n3 n X n2N c n 2 3 n X n2N b n3 n 1 2 X n2N c n3 n2C C=2 since (b n) and (c n) are sequences of 0’s and 2’sAs xwas arbitrary above, we obtain 0;1 A B Hence m(A B) 1, but nd Bare closed sets of measure zero. SOLUTIONS TO SECTION 1 5 Solution We have u x=u ss x u tt x= u s 2u t u xx=u sss x u stt x 2u sts x 2u ttt x= u ss 4u st 4u tt u xy=u sss y u stt y 2u sts y 2u ttt y= u ss 2u st u y=u ss y u tt y= u s u yy=u sss y u stt y= u ss Substituting these expressions into the given equation we nd. > µ î ð W h v µ v } ( o P o } µ Title math0b18w Author asalehigolsefidy Created Date 3/2/18 PM.

Written Homework 6 Solutions Section 310 30 Explain in terms of linear approximations or di erentials why the approximation is reasonable (101)6 ˇ106 Solution First start by finding the linear approximation of f(x) = x6 about 1 L(x) = f(a) f0(a) (x a) = f(1) f0(1) (x 1) = (1)6 6(1)5(x 1) = 1 6(x 1) Now to plug in the value to approximate, 101. SOLUTIONS 1 a F (A,B,C) = A’ B’ C’ A’ B’ C A B’ C’ A B’ C A B C’ A B C Distributive = A’B’ (C’ C) AB’ (C’ C) AB (C. Problem Set Problem Set #1 Math 5322, Fall 01 March 4, 02 ANSWERS.

B If A ≠ U and B ≠ U, fill in the blank with the most. HOMEWORK 8 SOLUTIONS MICHELLE BODNAR Problem 1 Suppose there exists a surjection f∶A→ B Recall that SBS ≤SAS if and only if there exists an. 2 PAULINHO TCHATCHATCHA One can also show by similar methods that (br)nc = (br)nc = (br)cn;.

And (bs)ma = (b s)ma = (b)am By de nition (br)n = (bm)1=nn = bm Similarly (bs)c = baThen (brbs)nc = (br)nc(bs)cn = (bm)c(ba)n = bmcbna = bmcna = (brs)nc QED (c) We want to show that br = supB(r) when ris rational First of all, we have to show that br is an upper. Math 2 Homework 4 Solutions Chapter 4 Problem 4 Suppose that x;y 2Z If x and y are odd, then xy is odd Proof Let x and y be odd integers By the de. 6612These Jordan matrices have eigenvalues 0, 0 0 0 They have two eigenvectors (one from each block) But the block sizes don’t match and they are not similar o 1 0 o J— 0 0 0 0 — 0 0 0 o) /o 1 0 o K—.

9 2Step 2 Calculate 2 2 s 1 &s 2s 1 = 396− 4 6 6−1 =24 28 6 1 6 18 68 2 2 2 = − − s = Step 21 Test to see if the two variances are homogeneous (ie H o 2. The determinant of the resulting upper triangular matrix is the product of the diagonals, and hence is 10 ( 4) (93=4) = 930 Since none of the row operations changed the determinant, 930 is the. Jan 27,  · Transcript Misc 3 Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C show that B = C In order to prove B = C, we should prove B is a subset of C ie B ⊂ C & C is a subset of B ie C ⊂ B Let x ∈ B ⇒ x ∈ A ∪ B ⇒ x ∈ A ∪ C ⇒ x ∈ A or x ∈ C (Since B ⊂ A ∪ B, all elements of B are in A ∪ B) (Given A ∪ B = A ∪ C) Taking x ∈ A x ∈ A.

4 1222 (d) Prove that f(f−1(B)) = B for all B ⊆ Y iff f is surjective Proof =⇒ Let y ∈ Y arbitrary We have to show that there exists x ∈ X with f(x) = y Let B = {y} By assumption, f(f−1(B)) = B = {y}, so y ∈ f(f−1(B))By definition this means that there exists x ∈ f−1(B) with f(x) = y. Department of Mathematics CSUSB CNS. M545 Homework 3 Mark Blumstein September 23, 15 Problems from Strauss’ Partial Di erential Equations 1) (21 #9) Solve u xx 3u xt 24u tt= 0, u(x;0) = x, u t(x;0) = ex Proof We begin by identifying the operator which de nes this pde,.

Problem 7 Write a proof by contraposition to show that for any real number x;. §36 19 Let A and B be n×n matricies a) Show that AB = 0 if and only if the column space of B is a subspace of the nullspace of A b) Show that if AB = 0, then the sum of the ranks of A. MAT 111 – Practice Chapter 2 Test 1 Express the set using set – builder notation {spring, summer, winter, fall} {x x is a season of the year}.

Caltech Math 5c Spring 13 Proof (a) Let K1 be the splitting eld of f 2 Fx, and K2 the splitting eld of g 2 FxThen K1K2 contains the roots of both f and gTherefore K1K2 is the splitting eld of the polynomial h = fg over F (b) Let g(x) 2 Fx be an irreducible polynomial with a root in K1 \ K2This means that g has a root in K1 and also a root in K2Since both K1 and K2 are splitting. If x3 x > 0;. 6 (a) Use induction to show F 0F 1F 2 F n 1 = F n 2 (b) Use part (a) to show if m6= nthen gcd(F m;F n) = 1Hint Assume m.

MAA6616 COURSE NOTES FALL 12 1 ˙algebras Let X be a set, and let 2X denote the set of all subsets of X We write Ec for the complement of Ein X, and for E;F ˆX, write EnF =. Title 2nd discussion Author hmkha Created Date 4/7/ AM. Solutions to Assignment 1 (c) Show that for all x,y ∈ G, we have x1−ny1−n = (xy)1−nUse this to deduce that xn−1yn = ynxn−1 (d) Conclude from the above that the set of elements of G of the form xn(n−1) generates a commutative subgroup of G.

Solutions for homework assignment #2 Problem 1 Show that the equation ∂u ∂t = k ∂2u ∂x2 Q(u,x,t) is linear if Q(u,x,t) = α(x,t)uβ(x,t) and in addition homogeneous if β(x,t) = 0. Title Topology of Fr Author scllo Created Date 4/9/ AM. 41 The derivative 43 Example 49 Define f R → R by f(x) = x2 sin(1/x) if x ̸= 0, 0 if x = 0 Then f is differentiable on R (See Figure 1) It follows from the product and chain rules proved below that f is differentiable at x ̸= 0 with derivative f′(x) = 2xsin 1 x −cos 1 x Moreover, f is differentiable at 0 with f′(0) = 0, since lim.

Introduction to Formal Language, Fall 16 Due 21Apr16 (Thursday) Homework 4 Solutions Instructor Prof WenGuey Tzeng Scribe YiRuei Chen. Suppose n(U) = w, n(A) = x, n(B) = y, and n(A ∪ B) = z a Why must x be less than or equal to z?. DCP3122 Introduction to Formal Languages, Spring 15 2Apr15 Homework 2 Solution Instructor Prof WenGuey Tzeng 1 Find all strings in L((a b) b(a ab) ) of length less than four.

CS/Math 240 Intro to Discrete Math 2/10/11 Solutions to Homework 2 Instructor Dieter van Melkebeek Problem 1 (a) First consider the case where the sets A and B are disjoint. HOMEWORK #10 { MATH 435 2 choose a= 2 1 n1 2R We notice that 2 is a root of the polynomial f(x) = xn1 2, which is irreducible in Qx by Eisenstein Thus ais algebraic and so a2K. Q Find the OTHER ENDPOINT of the segment with the endpoint of (4, 5) and midpoint of (7, 9).

Math 7350 Selected HW solutions Page 3 of 30 Given s>0, let A s be the atlas obtained from A0by replacing (V;. E) Æ Í {x} True – the empty set is a subset of every set f) Æ Î {x} False, the only element of the set {x} is x 15 Let A Í B and B Í C Show that A Í C By definition 4, from the first statement we have " x (x Î A ® x Î B) " x (x Î B ® x Î C) " x (x Î A ® x Î C). Thus, a plot of ln(y) versus x will yield a straight line with a slope of b 1 and an intercept of ln(a 1)The second equation is linearized by taking its logarithm to give.

MATH 351 Solutions # 7 1 Suppose that X is a normal random variable with parameters µ = 1 and σ2 = 9 (a) Find P{−2 ≤ X ≤ 1} Solution Since X ∼ N(1,9), we we have (X − 1)/3 is standard. Then x > 0 Solution Suppose x • 0 Then x3 • 0;. Show (AB) u (BC) = (A u B) (B n C) Proof First we consider the LHS (AB) u (BC) = (A n B c) u (B n C c) = (A u B) n (A u C c) n (B c u B) n (B c u C c) = (A u B.

Answer To show that this theorem in fact is a general property of products of metric spaces, I will write jx k y kjas d(x k;y k) We easily see that d euc(x;y) 0, and that d euc(x;y) = 0 if and only if x = yThe triangle inequality is. 47 Suppose B is countable and a correspondence f N B existsWe construct x in B that is not paired with anything in N Let x=x1x2 Let xi=0 if f(i)i=1, and xi=1 if f(i)i=0 where f(i)i is the ith bit of f(i)Therefore, we ensure that x is not f(i) for any i because it differs from f(i) in the ith symbol, and a contradiction occurs 410 Show that A is decidable, where. Whence x3 x • 0 ¥ Problem 8 A circle has center (2;4) (a) Prove that (¡1;5) and (5;1) are not both on the circle (b) Prove that if the radius is less than 5, then the circle does not intersect the line y = x¡6 Solution (a) It suffices to.

5 12 Evaluate the following expressions without using a calculator 12a arccos @ 5 6 A L 12b sec @arcsin 8 9 A L 13 An airplane flying at an altitude of 6 miles is on a flight path that passes directly over an observer. A E2M b E= VnN 1, where V is a G set and (N 1) = 0 c E= HN 2, where His an F ˙ set and (N 2) = 0 Here is a LebesgueStieltjes measure on Rand M is its domain (the measurable sets, where is the outer measure used in the construction of ).