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: _I. Let H and K be Hilbert spaces and A E B(H, K). Show that A is compact if = and only if A* A is…

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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 W = } t s,t e R -S - 31 5) Show that -6 is in W+. 3 6) Find a basis for W+.

A: given W=s+3tt-s-t  , s t ∈R

Q: 7) Let V and W be vector spaces over a field F. Recall L(V,W) denotes the set of all linear…

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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: Let B = be basis for R. Find the dual basis B* of B

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Q: Find a basis for the span of {[1, 3]", [-1, 3]T, [1, 4]T , [2, 1]T}

A: We have to find a basis for the span of   [1, 3]T, [-1, 3]T, [1, 4]T, [2, 1]T .

Q: Determine whether the set of polynomials 10. i (I+ x2, x + X2, 2 t x} is a basis forr Pe (TR).

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

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Q: Find the degree and a basis of the extension field Q(v3, v7) Q(V3 + v7). over

A: This question is from field theory's topic extension of field. Solution is given below.

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: Find a basis for span(sin2x, cos2x, cos 2x) in F

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Q: f e 17. Show that the set of all points in R2 lying on a line is a vector space with respect to the…

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

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Q: 3. [6 points] Consider the vector space V = P, over the field of real numbers F = R. Is the vector 4…

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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: 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: Consider the vector space V = P over the field of real numbers F = R. Is the vector 4 – x – x 2 in…

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Q: 2. Consider the basis * B = (1+*+**, x +r, 2). %3D for P2. Let p(x) = -1 + x + 2r?. Then [p(x)]R =…

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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: 4. Find a basis for the span of {[1, 2, 1]" , [3, 1, –1]", [1, –3, –3]"}

A: Given: uvw = 121, 31-1, 1-3-3 To find: The basis for the span. Concept Used: To find a basis for the…

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: Q1. Let V be a vector space over a field (F, +, .), let S={ vi, v2, V3 } be a linearly independent…

A: Let V be vector space over a field (F, +, .).

Q: Problem 3. Suppose the vector space V = P3, given the following four polynomials 1+13, 1+t, -t+2t2,…

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Q: The set W={ 2х х —2х `); x E R} is Select one: O a. Closed under both vector addition and scalar…

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Q: Let H be a Hilbert space and A E B(H). Show that (ker A*)+. (a) Image A = (b) ker A = (Image A*)+.

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