1) Let ACP(R), let f: R → R, and set A' = {0 CR : f-¹(O) E A}. Prove that A' is a o-algebra. Assume A is a 6-algebra !!!

Q: Suppose L: R² R³ is defined by L the given bases. [L(u)]s, = [L(u)]B₂ = ~Q[*^~-{]}]}----(HH:)…

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Q: Show that for any two finite-dimensional L-representations V and W of L: (a) Hom (V, k) ~V* as…

A: To show that for any tow finite-dimensional L-representation V and W of L HomkV, W≃V*⊗W as L-module

Q: Let F(a) be the field described in Exercise 8. Show that a2 and a2 + a are zeros of x3 + x + 1.

A: Given: Fa is a field. To show: a2 and a2+a are the zeros of x3+x+1.

Q: Let F = {0, 1} be the field with two elements, and let · denote the standard dot product in F100,…

A: Given: F=0,1 is a field with two elements. "·" denotes the standard dot product in F100.…

Q: Let f R2 R² be an isomorphism where : → ƒ ( [¹³]) = [()] and ƒ ([×4]) = }}] Find(G)

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Q: Prove that the following does not define an inner product R?: ((a, b), (c, d)) = ac – bd. on

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Q: Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves…

A: We have the function F:ℝn→ℝn that preserves the norm · This implies that this function satisfies all…

Q: In the frieze group F7, show that zxz = x-1.

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Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: True or False: (p, q) = p(1)q(1)+p(2)q(2) +p(3)q(3) defines an inner product on P2(R).

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Q: Use membership table to prove (AunionB) – C = (A – C) union (B - C).

A: We need to prove the following statement using membership table. A∪B-C=A-C∪B-C

Q: Let V be an n-dimensional vector space over a field F, where n ≥ 1. Prove that L(V, F) is isomorphic…

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Q: Please find all automorphisms. Thanks.

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Q: Can you help me explain why T is onto if Rank(T)=2 ?

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