a. Show that the field Q(vZ. v3) = (a+byZ +cv3+ dvZ3: a, b,c, d e Q) is a finite %3D extension of Q. Find its degree.

Q: Let d be a positive integer. Prove that Q[Vd] = a, b E Q} is a field. {a + bVā |

A:

Q: If the vector fieldv [z, y, 2mx+z]is irrotational, the mis equal to OA. 2 O B. 1 OC 1 O D. None of…

A: Correct option (B) 1/2

Q: If F is a field then F[x,y] is not a P.I.D

A:

Q: \Let X be Hilbert space over field F, and let R, S, T = S(X), 2 > 0: If S≤T then S+R≤T+R. (2) If S≤T…

A: Given : Let X be a Hilbert space over field F Let R,S,T∈SX , λ>0

Q: Let F = {a + bi : a, b e Q}, where i? = – 1. Show that F is a field.

A: Given F=a+bi:a,b∈Q, where i2=-1. We have to show that F is a field. First of all we define addition…

Q: 4. Is M2(R) a field? Justify. -{[i !] 5. Show that T = | |x, y, z €R is a subring of M2(R).

A: Introduction: A ring is another important algebraic structure under two binary operations. These…

Q: e) x·y= 0 iff x = 0 or y= 0. ) x<у iff — у < -х.

A:

Q: Suppose f(x, y) = x² + 2æy+ y². Which field plot is Vf? Justify your answer. 鹿

A:

Q: 2. Write and verify (i.e compute both sides of the equality) Green's Theorem for the field [P(x, y),…

A: Green's theorem states that, Let R be a simply connected region with smooth boundary C, oriented…

Q: Let v = [x2-y?, x2 +y², z] be a vector field. 1- Compute Curl(v).

A:

Q: Let v = [x2-y, x²+y², z] be a vector field. 1- Compute Curl(v). 2- Find the condition on x, y, and z…

A: Given vector field is  V = [ x^2-y^2 , x^2 + y^2 ,z]

Q: Determine if R is a field an integral domain a unital ring for R=Z/15Z, R=Z/16Z, R=Z/17Z

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Q: Let TE L(V, W), where V and W are finite-dimensional. Prove that rk(7) rk(7*).

A:

Q: If V is a vector space of dimension n over the field Zp, how manyelements are in V?

A:

Q: Prove that the fields Q(257) and Q(V257i) Here, i = V-1. isomorphic but not equal. are

A: Here , Q is a set of rational number.  So , here we will prove that the given fields are isomorphic.…

Q: (b) Prove that Q(Z[V3]) ~ Q[V3] where Z[v3] := {a+ bv3] a,b € Z} and Q(Z[/3)) is the field of…

A: Fraction field of a ring: Let R be an integral domain. The field of fraction Q is defined as, Q=ab:…

Q: 4. Consider the real vector space R?. For every x = (11,12) E R?, let ||x|| = |11|+ |r2l. a) Show…

A:

Q: (a) Determine whether the field F(r, y, 2) = (-2, 2, -2az - 62) is conservative or not.

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: 4. Show that the norm on (° given by I| (21, ..., In...) |lo= sup{|7;| : j =1,2, ...} %3D can not be…

A:

Q: a) Let R- (a+b VE: a, be Q. Prove that R is a field.

A: Since the second question is independent of the first question, as per the guidelines I am answering…

Q: 4. Is M₂ (R) a field? Justify. I 5. Show that T = = { [ ² ] 12,9₁² € R} i 2 is a subring of M₂ (R).

A: Subring of a ring

Q: (ii) Hence, use your answer obtained from Part b(i) to find the work done by the following field:…

A:

Q: If F is the vector field of this example, show that F. dr = 0 for every simple closed path that does…

A: Given the vector field F(x,y)=P,Q=-yx2+y2,xx2+y2. And from the example above, we have ∂Q∂x-∂P∂y=0…

Q: Let C:z-1 in the counterclockwise direction and / = . Lnzdz, J = , zdz, and K = $. cotzdz. For which…

A:

Q: | Given that U is a function of 2, yand z and A a vector field, prove that: V x (UA) = (VU) ×A +…

A:

Q: If R is a U.F.D., then R[x, y] is a U.F.D. If F is a field, then F [x, y] is à U.F.D.

A:

Q: Let u,v,w be linearly independent over a field F.  Show that{u+v,u−2v,u−v−2w}is also linearly…

A:

Q: If R is a U.F.D., then R [x, y] is a U.F.D. If F is a field, then F[x, y] is a U.F.D.

A:

Q: If F = Za[z] (z²–cz+1) find the value of c for which F will be a field? %3D

A: F is field iff x2-cx+1 is irreducible over f.

Q: Let {u, v, w} be linearly independent over a field F. Show that {u+v, u–2v, u-v-2w} is also linearly…

A: The given problem is to show the vectors are linearly independent , we can general laws to show…

Q: 5. Show that there is no vector field G such that: V x G = 2xi + 3yzj – xz?k.

A: Given that ∇×G→=2xi^+3yzj^−xz2k^. We have to show that there is no vector field G→ such that…

Q: Let Q be the field of rational numbers, then show that e(vZ, v3) = Q(vZ + J3).

A:

Q: In Exercises 13-26, a vector field Fand a closed curve C, enclos- ing a region R, are given. Verify…

A: The given function is: F=&lt;y2,x2&gt;; C is a triangle with vertices (0,0), (1,0) ,(1,1). To…

Q: Given an n xn-matriz A over a field K ry column vector b, there is a unique column v lution to Ar=0…

A:

Q: Let R be a UFD and fx) in REX) So, 0+a € IR.. Show that claf)~ ac(f) a nonzero polynamial

A:

Q: State whether the following statement is true or false in a vector space over a field F: V It a.BEF…

A:

Q: 11. Describe as many equivalent conditions as possible for a given vector field F is a conservative…

A: Given : F→ is conservative.

Q: Let v = [x2 -y, x²+y², z] be a vector field. 1-Compute Curl(v). 2- Find the condition on x, y, and z…

A: We have to find curl

Q: Determine whether T={(a, -a) | a EZ} is a subring of ZxZ-

A:

Q: For every vector field F on R³, curl(divF) = 0

A: Explanation of the answer is as follows

Q: If R is a U.F.D., then R[x, y] is a U.F.D. If F is a field, then F [x, y] is a U.F.D.

A:

Q: V be a finite dimensional vector spac field F, then there is a natural isom "onto V**. V**. |

A:

Q: Show that if k is a scalar and A is n × n, then det(k A) = kn det(A).

A: Show that if k is a scalar and A is n×n, then det(kA)= kndet(A)

Q: Show that if B is m × n, then BTB is positive semidefinite; and if B is n × n and invertible, then…

A:

Q: 1. Suppose that (R × R, +, .) is a field where (x, y) + (z, w) = (x + z, y + w) and (x, y). (z, w) =…

A: I have proved (1, 0) is the identity element and the checked by option one by one

Q: Let F be an ordered field and x,y,z ∈ F. Prove: If x > 0 and y xz

A: Given : x&gt;0, y&lt;z   To prove : xy&gt;xz

Q: Use the field norm to show: b) -1+v-3 is a unit in Z [1+ -3 ] 2 2

A:

Q: Let I and J be ideals of R. Prove that if I ∩ J = {0}, then rs = 0 for r e I and s e J.

A: Given- I and J are ideals of R. To prove- If I∩J={0} then rs=0 for r ∈ I and s ∈ J

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.