(8) Let u = (u1, U2, U3) and v = (v1, v2, v3). Define (u, v) = max{|u1 v1|, |u2v2l, |U3V3|} Determine if (u, v) is an inner product on R³. If it is, verify that the inner product ax- ioms hold. If it is not, list the axioms that do not hold and provide concrete example(s) to demonstrate that the axiom fails.

Q: Determine if the function defines an inner product on R², where u = (u, v) = ₁V₁ satisfies (u, v) =…

A:

Q: If A is diagonalizable and B is diagonalizable, must it be true that AB is diagonalizable? Give a…

A: If A is diagonalizable and B is diagonalizable then it is not necessary for AB to be diagonalizable.…

Q: Let (u, v) be the inner product on R² generated by A = [ 6 -1 and let u = (3, −5), v = (7,4). Find…

A: We have to find <u, v>

Q: Show that the polynomials in the set do not form an orthonormal set.

A: Given, The inner product: Polynomial set:

Q: Let ((u,,u2,U3),(V1,V2,V3))= 2u, V1 + 3u2V2+ U3V3 be an inner product on R. Determine ||(1, – 1, -…

A: Explanation of the answer is as follows

Q: Show that X is Hausdorff if and only if the diagonal △ = {(x, x)|x ∈ X} is closed in X × X.

A:

Q: Show that R is not compact.

A:

Q: 1. Consider the matrices A₁ 4 = ( 1 ) - 1 3 0 A₂ 42 = (3-2) ₁ 4₁ = ( 2² -2²), , A3 0 -2 These…

A:

Q: Let Consider the complex inner product space C³ with the usual inner product (u, v) = ₁V₁ + U₂ V₂ +…

A: Inner products space

Q: 6 Let (u, v) be the inner product on R² generated by A = - [ -1 and let u = (3, −5), v = (7,4). Find…

A: We have to find the .

Q: Find projvu and projuv. Use the Euclidean inner product. u = (7, -5, 1), v = (1, -1, 0) (a) projvu…

A: The given vectors and .We have to find the values of the projection,a) b) .

Q: Prov that Let A be a fuzzy ideal of R, anand by Lemma- be fuzzy singletons. Then (a.b) no A = anobu…

A: To prove that (a⋅b)h ⋅A=ah ⋅bh ⋅A given A is a fuzzy ideal of R and ah , bh are fuzzy singletons, we…

Q: True or False: (p, q) = pq defines an inner product on P(R).

A:

Q: 5. If X any Y are sets, then X-Y= {r€ X: 1¢Y} If X is open set and Y is closed set. Prove that (a)…

A:

Q: Let ((u1,u2,U3),(V1.V2,V3))=2u1 V1 + 3U2V2+ U3V3 be an inner product on R3. Determine ((1, – 1, –…

A: Given problem is :

Q: 2 2 6. State why = u,₁²v₁² + u₁₂²v₂² is NOT an inner product space for 2 u = (U₁₁U₂) and V = (V₁, V₂…

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Q: Explain why is not an inner product for u =(u1, u2) and v =(v1, v2) in R2 when = u1u2 + v1v2 using…

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Q: Find projvu and projuv. Use the Euclidean inner product. u = (-1, 2, 1), v = (1, 2, 1) (a) projvu…

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Q: Determine whether each product is an inner product space in R3. If not, explain why. 1) = u1v1 +…

A:

Q: Find projyu and projuv. Use the Euclidean inner product. u = (-1, 2, 1), v = (1, 2, –1) (a) projyu…

A: Given u = (-1, 2, 1) and v = (1, 2, -1)

Q: 2. Determine if the function (u, v) = µ₁ V₂ + U2V1, where ủ = (u₁, U₂) and ở = (V₁, V₂) defines an…

A: