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Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: Given the bases B = {u,,u2} and B'={u'1, u'2} for R², where and u'1= The transition mat rix Pg-8'…

A: Given bases  are B=u1,u2 and B'=u1',u2'  where u1=10,u2=01 and u1'=12,u2'=11 To find the transition…

Q: Let J={a: aER}, then I is Borel algebra.

A: False Statement' Detailed Explanation

Q: Let z be a complex mumber o B19) s and c be any then So BlE)dz+a fle) dz isequi contou any funckon…

A:

Q: Prove Theorem A(rB) = r (AB) = (rA)B

A: Given: A(rB) = r (AB) = (rA)B

Q: Let E={(r,0,z) : os es T, OS zs 1, 5Vzs rs 5. Compute

A: Introduction: The triple integral ∫∫∫Ef(x,y,z)dV with the region E=(x,y,z):a≤x≤b, c≤y≤d, e≤z≤f can…

Q: Prove VZEC, V2|z|>|Re z| +|Im z| + |Im z| Vze C,

A:

Q: Let A E Mnxn(R), show that it is possible to express A as the product, A = PQ where P,Q € Mnxn(R),…

A:

Q: Let A € M₂ (R) be given. (a) If AA¹ = 0n, prove that A = On. Does this hold if A € M₂(C)? (b)…

A:

Q: prove that 01a| = 0 => a=o @ 1-al-lal @lak b ⇒-b<a<b @la-bla lal-lb/

A:

Q: Exercise 2.5 Prove that if m" (AAB) = 0, then m* (A) = m* (B). Hint Note that AC BU(AAB).

A:

Q: (b) Let 2a13 2a12 a11+a21 a12+ a22 a13+a23 2a11 a11 a12 a13 A = a21 a22 a23 and B : a31 a32 a33 a32…

A: The determinant is a scalar value that is a function of the entries of a square matrix. It allows…

Q: 11. Is the following decomposition in BCNF (hint: think lossless join) R=(ABCDE)|R1(AB) | R2(BC)|…

A: Given: To explain the given statement as follows,

Q: 2. Prove that there are no rectangles in S2. More precisely, prove that if A, B, C, D E S² form a…

A: Given: ∠A=∠B=∠C=∠D=π2

Q: Prove that c = cn. i=1

A:

Q: Find all isometries on C which map 1 to 2 and 1+ i to 1.

A: Answer is provided below  If you have any doubts,feel free to post the same again

Q: Given that n ≥ 2 , x ∈ Rn  Prove that there exists a y ∈ Rn in where y ≠ 0, and xy = 0

A:

Q: Show that A is skew symmetric if and only if ai j = −aji for all i, j.

A: Suppose A be any square matrix of order n such that A=a11a12⋯⋯a1na21a22⋯⋯a2n⋯⋯⋯⋯⋯an1an2⋯⋯ann First,…

Q: Given the bases B = {u1, u2} and B' = {u'1, u'2} for R?, where and u'1= The transition mat rix Pg-B'…

A:

Q: Part Ca):- a*t a e CCR) for all ae R sthen prove that R is commutative.

A:

Q: m is the mid point of AB find a Of AM

A: Equating the given relation gives the value of x . Then substituting the value of x in the…

Q: Evaluate det(A) by using expansion along a row or col- umn, given 1 2 1 3 4 A = -2 -1 1 1 -1

A: Given matrix is, A=01201345-20-111-100

Q: Evaluate det(A) by using expansion along a row or column, given

A: Given:         A = 01201345-20-111-100   To find :     det(A) using expansion along row or column

Q: 3. prove that: det (AB) = det (A) det (B) a - (; ), » - ( ) 2 -1 7 2 B = 2 в 3 %3D

A:

Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

A:

Q: Find the projection ofv onto u

A: We have given two vectors v=-24 and u=-11 We have to find the projection of the vector v onto u.

Q: Given the bases B= {u,,u2} and B'= {u'1, u'2} for R, where U2 and The transition mat rix Pg-g from B…

A: We first find the coordinate vectors of u_1 and u_2 relative to basis B' ,then we will form the…

Q: Let A, B, and C e M,(C) such that ВС - СВ %3D А"А prove that BC = CB.

A:

Q: Prove that there do not exist a, b ∈ Z satisfying 54a − 39b = 2.

A: Solution:

Q: Find all units of . ZP . Zn . Rn x n . {0,3,6,9} as a subring of Z12

A:

Q: Let a,b E R with a # 0, b # 0. Prove that (ab)¯1 = a°1b´1. %3D

A:

Q: Let T={[a,0) : aER}, then T is not Borel algebra

A: Given a statement

Q: 1. Let s = (001 001 001...) t = (01 01 01...) r = (10 10 10...). %3D Compute: a. d[s, t] b. d[t, r]…

A:

Q: Prove that D24 and S4 are not isomorphic

A: Since D24 ={a,b: a 12 =b2 =e (identity) and ab=ba-1 } This implies that D24  has 12 reflection and…

Q: (a) Let r = ri+ yj + zk and let r = ||r||. Prove that - (1) a

A:

Q: In AABC, AC > AB. The biscctor AD of ZBAC meets BC at D. Prove that Z ADC is obtuse. B C

A: In ∆ABC, AC&gt;AB. The bisector AD of ∠BAC meets BC at D.To prove that ∠ADC is obtuse.

Q: Prove Theorem . k(AB) = {kA)B = A{kB)

A: Proof: Given:   Let A and B are matrices of the same order and let k be the scalar.  To prove:…

Q: E.

A: By using the figure to answer the following questions by YES or NO.1 Are E, I, and A coplaner.2 Are…

Q: 19. Given: OABCD; BE | MD: M is the midpoint of AB. Prove: DE = BC %3D E D. A M В

A: We have the given figure as, Here, □ABCD is a parallelogram and in figure BE∥MD. M is the mid point…

Q: (a) Let r(t) = ti+tej+fk. Find ((1),4 × (?),1) - (1)a tuj

A: First we find out first derivative of r(t) with respect to 't' then find out 2nd derivative and then…

Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

Q: Show khat the glosp brjections frem f5 faom f5 whan f rafiell iisomaphi- with

A:

Q: Exercise 1 AD bisects BE AB || DE AABC = ADEC Given: Prove: A E

A: As per guidelines we can solve the first question only. If you need more answer please repost it.…

Q: Find the angle between x and x4 in C[0,1] with its standard inner product.

A:

Q: Let 1 an(r) : u(reie)e ino de %3D Prove that:

A: .

Q: Prove that (i) Res (tan z, r/2) = - 1

A: Here,we have to prove that Res(tanz,pi/2)=-1

Q: 1. Given that Zo of z is A. i B. 1

A:

Q: A subset W of R° is given by = {(x,y, z) | 2x = y = 2z}. (i) Write down the distinct elements of W.

A:

Q: Let J={(a,∞): a€R}, then T is Borel algebra. صواب İhi

A: True

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.