6. a) State the Fundamental Theorem of Galois theory. b) Outline the proof of (a).

Q: Let V be a complex vector space and f a form on V

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Q: 1. Let n e N and let V be the set of polynomials in the indeterminate x with coefficients in R of…

A: According to our guidelines we are only supposed to answer the first three subparts of the question.…

Q: 8. Prove the completeness relation: if |1), .. n) is an orthonormal basis, then %3D

A: Use the Dirac notations.

Q: 7. Theorem : Let V and W be vector spaces over a field F. Let {v1,.., vn} be a basis of V and let…

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Q: 1.3.1 Determine whether or not the given vector spaces V and W are isomorphic. *(a) V=R, W= M2,3(R)…

A: The two vector spaces are isomorphic if there exists a bijection between them T:U→V such that…

Q: 4. Let Let 2 3 -2 5 S= -3-2 DO is a subset of R. Find an orthonormal basis for S. (Take dot product…

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Q: a) Let a = V2+ v2 i) | Determine the minimal polynomial of a over Q. ii) | Write in terms of a…

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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: Let V and W be vector spaces over a field F. Prove that V is isomorphic to W if and only if dim(V) =…

A: Let V and W be vector spaces over a field F. We prove that V is isomorphic to W if and only if…

Q: Let V and V' be finite dimensional vector spaces over a field F and B = {b1, ba,., b,} and B' =…

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Q: 2. Let G by the symmetric group S(n) and V the complex vector space with basis {v1, v2,..., vn}. For…

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Q: 1 0 -2 Let T be defined by T(x) = Ax where A =-2 1 3 -2 -5 a) Find T b) Find a vector x whose image…

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Q: 2. Let V be a vector space of dimension n and let B = {v1,..., Un} be a basis of V. (a) Prove that…

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Q: Given A i|1){1|+1)(2|+ 2|2)(1| – |2){3| with basis {|1), |2), |3)} in C³, (a) Find At. (b) Show if A…

A: The complete solutions are given below

Q: Let V be the vector space of functions spanned by Nπ B = {b₁ = 5 sin x + 7 cos x, b₂ = 3 sin x + 9…

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Q: 10. Suppose v1, ..., Vķ are nonzero vectors with the property that v; · v; = 0 whenever i + j. Prove…

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Q: (6) Suppose that T is a linear operator on a 2 dimensional vector space V and that T+ a I for any a…

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Q: 1. Let V be a vector space. Let v be a vector in V and c a scalar. Proof the following statements.…

A: (a) For each vector v, the negative vector -v is defined as            v + (-v) = (-v) + v = 0.…

Q: Exercise 13. Let x € V and c; be the Fourier coefficient of x with respect to v; , and let {v,,v2,…

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Q: Which of the following statements are/is true? (i) V1, V2 is a basis for span of S 0. (ii) The…

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Q: 3) Let B basis for a topological space T on X and B' is a family of sub sets of cuch that 6 C R'CT.…

A: Topological Space: Suppose  is a non-empty set. A subset  of the power set of  is said to be a…

Q: Consider P2(R) with its basis B = (x, x + 1, x² –- 1). Find [x2 + 3x + 5]B.

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Q: 12) Prove that if a and b are constants and f and g are scalar fields, then V(af bg) aVf bVg

A: Take the scaler fields:

Q: (a) Let A = {₁,...Un} be a basis for a finite dimensional real vector space V. Define the…

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Q: Let {₁,..., wk, Wk+1,...,n} be a basis of R" and c₁,...,CER be nonzero scalars. (a) Prove that S =…

A: Note that we used one fact in this prove which state that if B is basis of V and C is any linearly…

Q: 2. Consider the vector space F = ((1, z2) | 1, 72 E F}. How many TE L(F) are isomorphisms? List them…

A: Consider the vector space F22=x1,x2: x1,x2∈F2. We have to list all isomorphism s.t T∈LF22. We have,…

Q: Let V, W, and Z be vector spaces, and let T: V → W and U: W → Z be linear. (a) Prove that if UT is…

A: Note: We will be solving part (c) only as asked.Let  and  are vector spaces.Let  and  be linear.We…

Q: Let R be a relation on the integers Z such that (a,b) = R if and only if ab = 0.

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Q: a) [The set v = U2 U1- V2. [2].110 b) a basis for R². The set S = c) The set S = can If v E span…

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Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V x V → V…

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Q: 7. Let V be an n-dimensional vector space over the field F and let B = {V₁, V2, ..., Vn} be an…

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Q: Let V be n-dimensional vector space over c en for any n scalars ₁, ... , E F there-

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Q: EV and V' be vector spaces over a field I xists a unique linear transformation, t: V at

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Q: 3) Let ß basis for a topological space T on X and B' is a family of sub sets of X such that B C…

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Q: Define F,[x] = {f(x) € F[x]]deg ƒ (x) sn}. One can show F„[x] is a vector space. Let U = {f(x) €…

A: here we can use span of a set and linear combination.

Q: 2: Consider B= a) Show that B is a basis for R³.

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Q: 0. Let V be the space spanned by v₁ = V3 = cos 2x. (a) Show that S = cos² x, V₂ = (b) Find a basis…

A: Given that V be the space spanned by v1 = cos2x, v2 = sin2x, v3 = cos2x.a) We have to show that S =…

Q: (a) Let f: V > V be any linear map where V is a vector space of dimension n over a field K. Suppose…

A: Let f : V -->V be any linear map where V is a vector space of dimension n over a field K. Suppose…