Prove, that the vector Space Hat (n; F) with the "multiplikaton" A*B:= AB- BA is a F-algebra ( algebra over a field F). is such an algebra associative, commutative, unitary ?

Q: Let S be a finite set in a vector space V with the property that every x in V has a unique…

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Q: Prove that if V is a finite-dimensional vector space over a field F, then a subset {βi , β2 , . . .…

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Q: Let F be a finite field of order q. Let E be the splitting field of Xª – X +1 over

A: Let F be a field of order q Consider the polynomial fX=Xq-X+1∈FX We have to determine the order of E…

Q: Let V be a vector space over some field F and let L = {V₁, V₂, ..., Uk} CV be a finite linearly…

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Q: Show that the set  Fm×n of all  m×n  matrices over a field  F  is a vector space over   F with…

A: Given:- Fm×n: Set of all m×n matrices over field F . To prove: We have to show that the set Fm×n of…

Q: ) Let V be a vector space over any field F: (a) Prove that V is infinite dimensional iff there is a…

A: AS YOU ASKED MULTIPLE QUESTIONS AND EACH QUESTIONS CONTAINS DIFFERENT PROOFS SO ACCORDING TO THE…

Q: Suppose F is a field of characteristic p. Show that F can be regarded as a vector space over GF(p).…

A: Given F is a field of characteristic p. For k∈GFp, x∈F. For F to be a vector space the product is…

Q: Let U be a linear operator on a finite-dimensional inner product space V. If ||U(x)||= ||x||for all…

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Q: .1) (a) Define a vector field in 2-space with an example. Graph of the vector field F(r, y) = yi –…

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Q: Let V and V' be n and m dimensional vector spaces over a field FE T

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Q: Prove thatc "is a vector space over the field of real numbers as well, for any n

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

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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: Let V be a vector space over a field F, and let W be a subset of V. We say that W is closed under…

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Q: Give a basis for the indicated vector space over the field. Q(i) over Q

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

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Q: For each of the following vectors spaces over a field F , provide a basis and compute the resulting…

A: As per our guidelines we can answer only 1 question so kindly repost the remaining questions…

Q: Let V be a finite-dimensional vector space over a field F, and let T:V → V be a linear…

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Q: 1. Suppose U and V are n-dimensional vector spaces over the same field F. Prove that U and V are…

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Q: (x|y) = (a|x;)(y|r;). (x|x;)(y|x;). i=1

A: Consider V is a finite-dimensional inner product space over F. Let x1,x2,x3,....,xn be an…

Q: 17. Let V = {(a1, a2) : a1, a2 E F}, where F is a field. Define addition of elements of V…

A: We have to check whether the set V=a1, a2:a1, a2∈F is a vector space or not. The addition of vectors…

Q: Let V be a finite – dimensional vector space over the field Fand i 1. 11 c 1 1 :

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Q: Prove that for a vector space V over a field that does not have characteristic 2, the hypothesis…

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Q: Show that the field R(i) as a vector space over R has a basis { 1 , i } and is isomorphic to  R_2,…

A: The field ℝ (i) is defined as ℝ (i) = {a + bi| a, b ϵ ℝ}.That implies, ℝ (i) = {a × 1+ b × i | a, b…

Q: State whether the following statement is true or false in a vector space over a field E: " a,BEF and…

A: Given that , V is a vector space over a field F.

Q: Prove that ifV is a finite-dimensional vector space over a field F, then a subset (B, B2,.. Bn) of V…

A: First, show that if each vector in V can be uniquely expressed as a linear combination of the…

Q: (a) Obtain the splitting field of (x2 – 5) (x² – 7) over Q. Obtain the Galois group of this…

A: Find the attachment.

Q: If F is a vector field and there exist a closed curve C such that SF. dr = 0 then Fis conservative.

A: Since you have posted a multiple question according to guildlines I will solve first question for…

Q: If F is a vector field, and C' is an oriented piecewise smooth curve consisting of smooth curves C1…

A: Given: To find: Whether the given statement is true or false.

Q: If (e,} is an orthonormal basis for a Halbert space H, prove that || x ||2=£ | |² for all x in H.

A: An inner product space which is complete with respect to the norm coming out of the inner product is…

Q: Give an example of a vector space V over a field F, and of a linear transformation T: V –V that is:…

A: A transformation T:V→W is injective(one to one) if each element of V mapped on distinct element of…

Q: If is a vector field, then div is a vector field

A: FALSE the divergence is a scalar function.

Q: 8) a) Determine how many basis exist for the two-dimensional space F over the field F,.

A: As per our company guidelines, we can solve first question for you. If you want other question to be…

Q: Let S = {x1, x2..., xn} be a linearly independent subset of a vector space V over the field Z2. How…

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Q: Suppose that K is an extension field of F with a, B,y E K Prove that if {a, B, 7} is linearly…

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Q: State if the following statement is true or false. If V is a vector space over a field f and p(V) is…

A: This is the problem of linear transformation and vector space.

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=<y2,x2>; C is a triangle with vertices (0,0), (1,0) ,(1,1). To…

Q: Give an example of a vector space V over a field F, and of a linear transformation T : V –→ V that…

A: Let V be a vector space V over a field F and T:V→V To Prove : Surjective (onto) but not injective…

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

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Q: Let V be a vector space over a field F, and let W be a subset of V. We say that W is closed under…

A: This is a problem of vector space.

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

A: Given : F→ is conservative.

Q: 23. Prove that every finite-dimensional vector space V of dimension n over a field F is isomorphic…

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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: Give an example of a vector space V over a field F, and of a linear transformation T: V → V that is:…

A: Let V be a vector space V over a field F. and Linear Transformation T:V→V

Q: a Prove that the ring of matrices :a, b e Q> is a field. 2b a

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

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Q: Give an example of a vector space V over a field A a) injective but not surjective; b) surjective…

A: We need to provide a function that is injective but not surjective, and a function which is…