5. Let p be a prime number and let Fp denote the field with p elements, Fp = {0, 1,..., p-1}, equipped with the operations of arithmetic modulo p. Consider the action of SL2 (Fp) on F2 defined by A(V) := Av. Show that the orbit of the vector (1) is the set of all nonzero vectors in F2. How many elements belong to the orbit and stabiliser of this vector? Use these answers to calculate |SL2(Fp)|.

Q: Consider the vector space P, of polynomials with real coefficients of degree at most two together…

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Q: - c(13, 23, 21, 27, 37, 52, 34, 42, 31, 40, 54, 30, 15, 35, 19, 23, 22, 18) Fine the vector y by the…

A: year x 1951 13 1952 23 1953 21 1954 27 1955 37 1956 52 1957 34 1958 42 1959 31…

Q: Consider the vector field F(r, y, z) = (2ry³z*, 3x²y²zª, 4x²y³z³). (i) Show that F is a conservative…

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Q: Let {u1, let B be a basis for V. Let S vectors of {u1,·· , Un} with respect to basis B. Prove that ,…

A: Given V is a n- dimensional vector space. Let B be a basis for V. Let B=v1,v2,…,vn. Let u1, u2, …,…

Q: Show whether or not the set of vectors B = {U, V, W} where U = V = and W = 0 is a basis for R³.

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Q: Show that the set is an orthonormal basis for R3.

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Q: Let 7 = -27+ 7+2k and w = 3 7 +23- k. (a) Compute the vector projection of w onto , proj- w. (b)…

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Q: 1) Suppose Ris the set of real numbers, and we define addition (using the symbol ) as: x ® y =…

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Q: Show that λis a root of char(x) if and only if there exosts a non-zero vector v in V with L(v)=λv

A: Assume that λ is a root of charL(x) then | λIdv–L | = 0. It is known that, the system of Bx = 0 has…

Q: 13. Do the set of vectors S = { x²+2x-4, 6x² +7x+3, 2x²+x) a basis for P2? Refer to the definition…

A: The given set is;  S=(x2+2x-4, 6x2+7x+3,2x2+x).

Q: Find a splitting field S of the polynomial x^4-7x^2+10 over Q. Find [S:Q] and a basis for S/Q.

A: The given polynomial is x4-7x2+10 over the field of rational numbers ℚ.

Q: 2. Let F = Q, K = Q(V5), and E = K(i, 5+ i). (a) Regard K as a Q-vector space (i.e. with Q as the…

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Q: 3) V. (FF) = f (V. F ) + F •V ƒ. 4) V: (F +G) = V. F + v•Ğ. 5) V: (FxG) = Ġ- (V × F ) – F · (V × Ġ)

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Q: sider the ordered bases B= ((4, –9), (–3,7)) and C = ((4,3), (–2,4)) for the vector space R’ %3D

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Q: 7) The vectors. A) True 13 False 17 -19 + -34 38 17 2 form a basis for n

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Q: Determine

A: Length of all vectors is 1.

Q: Let Pn = {anxn + an-1xn-1 + . .  + a1x + a0 | each ai is a realnumber}. Is it possible to have a…

A: The polynomial space is defined as follows: Pn=anxn+an-1xn-1+⋯+a1x+a0  |ai∈ℝ Check whether is it…

Q: 2. Let V be the set R regarded as a vector space over the field Q with the usual operations. Show…

A: To show that the set ππk,k=0,1,2,... is linearly independent in the vector space V over the field Q…

Q: hich of the following sets of polynomials form a basis for P2? 1, p2 1- 5х, рз || (i) p1 х, р4 (i)…

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Q: REMOVE vectors from Ac (P,[R], +,). To obtain a BASE for (P.[R],+,): of the items 25 and 26 (Item…

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Q: 53. Consider the basis B of R² consisting of the vectors [3] [F]g = | 1 and We are told that i 93…

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Q: 8. Let H be the set of all vectors of the forms given below and either find a basis for the vector…

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Q: 5. Prove that if F, G are vector fields on R3, then curl(F + G) = curl(F) + curl(G).

A: Let the vector fields F and G are: F=Fxi+Fyj+FykG=Gxi+Gyj+Gzk Then,…

Q: Let A = -2 -2 5 5 -5 -2 Give a non-zero vector in the null space of A. -5 -2 -1 5 5 1 2

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Q: 5. P2 denotes the set of all polynomials of degree ≤ 2, in a single variable x, with real number…

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Q: Consider the following polynomials as vectors from the vector space P3 (R) (Definition 2.13 in the…

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Q: Let V denote the space of polynomials in x of degree ≤ 3 over a field K. Let v1 =1+x+x2 +x3, v2…

A: Introduction  We have given  the space of polynomials in x of degree ≤ 3 over a field K. Let v1…

Q: -10000- Show that the set spans R. Find a subset of these vectors that is a basis 3 2 3 for R3. 2.

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Q: Represent the vector ⟨16,8,−14⟩ in terms of the ordered basis B={⟨5,−2,−2⟩,⟨1,−4,2⟩,⟨−1,−5,3⟩}.

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Q: What is the dimension of P₂, the vector space of all polynomials of degree at most two?

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Q: ༤ས་ Let A = = [3 0 0 -3 -2 -1 A basis for the null space of A is { vector

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Q: Suppose f is the function defined by f(x) = Ax, where a. The domain of f is b. The codomain of f is…

A: Dear student, as per our guidelines we can answer only first three subqustions. So kindly resubmit…