MAA 3200 Midterm: MAA 3200 FIU Exam 113k

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turquoisegnu128

15 Feb 2019

School

FIU

Department

Mathematics - Analysis

Course

MAA 3200

Professor

All

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Prof. s. hudson: [20 pts] answer true or false. (p q) p q ( k z, 2k = 8) ( k z, 5k = 25) K z, (2k = 8 5k = 25) ((p q) p ) q is a contradiction. !x r, (x 1)(x 3)(x 5) = 0 |x 2| < 2. If n 7 is an integer, then there are positive integers k, j so that n = 2k + 3j. There is a set that is a member of every set. In the real numbers, x, y, w, wxy < w2. If a is a proper subset of then a = {17}: [15 pts] let a, b be sets with p (a b) = p (a) p (b) (as usual, p refers to the power set). Prove that a b or b a: [15 pts] prove that for all integers n 1, The next few problems are intended to be fairly easy and short.

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I have the answer for the following question. But some of theanswers are not correct.

Please do the following question by yourself and check which arethe wrong answers,please post your answers and compare to it.

Let r in and u =(1 1 0), v = (1 4 1), w = (r2 1 r2), b = (3+2r 5+12r 2r). For which values r in is the set {u. v, w} linearly independent? For which values r in is the vector b a linear combination of u. v and w? For which of these values of r can b be written as a linear combination of u. v and w in more than one way? The set 3 of all column vectors of length three, with real entries, is a vector space. Is the subset B = {(x y z) in 3 | xy + yz = 0} a subspace of 3? Justify your answer. Show that the set of all twice differentiable functions f : satisfying the differential equation sin(x) f"(x) + x2f(x) = 0 is a vector space with respect to the usual operations of addition of functions and multiplication by scalars. Here, f" denotes the second derivative of f. Let S be the following subset of the vector space 3 of all real polynomials p of degree at most 3: S = {p in 3 | p(l) = 0, p'(l) = 0}, where p' is the derivative of p. Determine whether S is a subspace of 3. Determine whether the polynomial q(x) = x - 2x2 + x3 is an element of S. u = (1 1 0), v = (1 4 1), w = (r2 1 r2) if {u, v, w} is linearly independent set 1(4r2 - 1)-1(r2 - r2) ne 0 4r2 ne 1 r ne plusmn1/2 this set is linearly independent set for values of e ne plusmn1/2 - - - - - - - (3+2r 5+12r 2r) = a1 (1 1 0) + a2 (1 4 1) + a3 (r2 1 r2) this forms system of equations in a1, a2, a3 if 4r2 - 1 ne 0 then system has unique solution if 4r2 - 1 = 0 and -2 - 4r = 0 rArr r = -1/2, the system has more than one solution if 4r2 - 1 ne 0 b can be written as linear combination of u,v,w as uniquely if r = -1/2 then b can be written as linear combination of u,v,w more than one way - - - - - - - - - (1 1 -1) in B and (1 0 1) in B (1 1 -1) + (1 0 1) = (2 1 0) not in B since 2times1+1times0 ne 0 so addition of vector is failed, this set is not forms subspace consider the following differential equation (sinx)f"(x) + x2f (x) = 0 each solution is continuous function the set of all solutions is subset of space of set of all continuous functions n from R to R let f,g are two solutions (sinx)f"(x) + x2f (x) = 0 (sinx)g"(x)+x2g(x) = 0 ccheck whether af+bg is solution or not (sinx)f"(x) + x2f(x) = 0 rArr (sinx) af"(x) + x2?f (x) = 0 (sinx)g"(x) + x2g (x) = 0 rArr (sinx)bg" (x) + x2bg (x) = 0 (sinx)(af"(x)+bg"(x)) +x2 (af (x) + bg(x)) = 0 af+bg is also solution to differential equation the set of all solutions to differential equation forms subspace - - - - - - - - - - consider the following subset of P3 {p in P3/.p(l)=0, p'(l) = 0} let p, q in S rArr p(l)= 0, p'(l) = 0 and q(l) = 0, q'(l) = 0 (ap + bq) (l) = ap (l) +b q(l) = a (0) + b (0) = 0 (ap + bq)'(1) = ap'(l) +bq'(1) = a (0) + b(0) = 0 p, q in S rArr ap +bq in S then S is subspace q(x) = x - 2x2 + x3 q(1) = 1 - 2 + 1 = 0 q'(x)=1 - 4x + 3x2 q'(l) = 1 - 4 + 3 = 0 q is element of S

ey table is seen below xeecions tor Chapter 1 12, Fori, b â¬ Î.el * b-aa (0 (D 14 ForVEQ r the operation defined in Example 1 on the wr John Sue Henry. P iDFor 15ine if is ass ociative by determining if the final six engu defined ang property (ii)of Theorem 1 . So no element i if the operntion debined in Example 1 of Emle if each element has an inverse Section 1.3 Groups iw comutatityn ild Chapter 3 where the Funda the exercises that follow etermining why they are not Write out the Cayley table for the gru(o)nud lensify thee element the 10 elements of the group Z 18 its operation uses ts in the first coordinnte and +2 in the secotu Ldessity the Zn and write out the Cayley table. Reeall th of each element does Theorem 1.14 tell us aont the entries in the Cayley table ol s xrewp that is set with exactly one element. A {n), will always form group since there 19. What nly one way to define an operation on the set. Be sure to shhow that the properi combination of the usual notice what set the rule is be of a group all hold. 21. We can create even create groupe with games! Consider four cups placed in a ssare patter table. If we have a penny in one of the cups there are four ways we can move it to t cup: Horizontally, Vertically, Diagonally, or Stay nehere it is. We will p, S. To define nn operation, consider two movements in a row, v.e.,m ove the penny as r tells us to, then after that move the penny as y instruets. the set of integers the set of real numbers n the set of positive integers. e set of positive integers. n the set of positive integers. on the set of integers e set of negative integers. we ple, HV D since if we first move it horizontally and then vertically nltogether e moved it diagonally. Create the Cayley table for this group, identify the identity and the inverse of cach element 22. Prove that the set A 0 under matrix multiplication. Is it abelian? 23. Verify that 3(R) is a group using the addition of funetions as defined in Example 13 e set of nonzero rational n he set of nonzero real num n on the set R (-1 Section 1.4 Basic Algebra in Groups 24. Prove the Cancellation Law (Theorem 1.20). Do not use any theorems that occur after it in the text ssociative, commutative, has roof or counterexample fo 25. Assume G is a group and a e G. Consider a fixed integer k and use PMI (Theorem 0.3) to prove that for all nEN, a aYou will need to consider cases here since k can be positive, negative, or o 26. Prove Theorem 1.23 (ii) for the case of n > 0 and m

These questions are asking about subrgoups, groups, isormorphic.

actually, after 2day, I got a midterm for abstract math test,and I got a sample of midterm test, so I hope to get a solution ofall these question. I think it is a lot, but I hope that pleaseshow me the answer and I want to know why these answer is this.

The reason why these problems are bunch of questions, so I willgive big pts for this questions. thanks.

you guys can upload by the picture.

(below question)

SAMPLE MIDTERM 1, MATH 120A

(1) 1. Write the denition of group.

2. Give an example of a group of order 2011.

3. Give an example of a non-abelian group with 12 elements.

(2) Let (G,*) and (H . ) be groups.

1. Dene what it means for G and H to be isomorphic.

2. Assume that G has exactly two solutions to the equation x^2 =e.

Prove, directly from the denition in (1), that if G and H areiso-

morphic, then H has exactly two solutions to the equation x^2 =e.

3. Prove or disprove: (R, +) is isomorphic (R+, . ). Here R+ ={x is subset of R : x>0}.

4. Prove or disprove: (R, +) is isomorphic (R*, . ). Here R* ={x is subset of R : x is not 0}

(3) Let G be the group Z_20, with addition modulo 20.

1. Draw the subgroup diagram of G, giving a generator for eachsubgroup

2. Find the subgroup H =< 16 > of G.

3. Find a subgroup K of U_20 isomorphic to H.

(4) Let (G, . ) be a group and let H be a subgroup of H. Fix anelement a of

G and let K = { aha^-1 : h is subset of H}

1. Prove that K is a subgroup of G.

2. Prove that the map psi : H -> K, h -> aha^-1 isan isomorphism of groups.

(5) Let (G, . ) be a group and let a be an element of G.

1. Dene the order of a.

2. Give an example of an element of order 4 in GL(2,R).

3. Find the order of 16 in Z_20.

MAA 3200 Midterm: MAA 3200 FIU Exam 113k

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