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

Q: Use the functions f and g in C[−1, 1] to find d(f, g) for the inner product 1 (f₁ g) = [ { f(x 1…

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Q: Give an example of each of the following, and show that your example satisfies the "highlighted"…

A: For the solution follow the next steps.

Q: If f(x) and g(x) are arbitrary polynomials of degree at most 1, then the mapping (f,g) = f(-2)g(-2)…

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Q: Let a, b, and c be distinct real numbers. Show that…

A: Consider the equation: We have to check whether it is a zero polynomial, symmetric and bi-linear or…

Q: Prove or disprove: there is an inner product on R² such that the associated norm is given by ||(x,…

A: We need different  property/condition that a  norm  must satisfy.   One of the condition is given…

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

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Q: Verify that the operation {x, y} = x1y1 − x1y2 − x2y1 + 3x2y2 where x = (x1, x2) and y = (y1, y2) is…

A: Given : The operation on ℝ2 is defined as  x, y = x1y1 - x1y2 - x2y1 + 3x2y2 ,  where x = (x1 , x2)…

Q: Prove that (Z(sqrt(2)), +, ×) is an integral domain.

A: We know that , ℤ2 = a + b2 / a , b∈ℤ . Clearly , ℤ2 forms a ring. We know that , a commutative ring…

Q: 5 dx 8x (x+1)? 1

A: First evaluate indefinate integral

Q: Use the inner product (p, q) = aobo + a₁b₁ + a₂b₂ to find (p, q), ||pl|, |la||, and d(p, q) for the…

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Q: Show why the product {[x1, x2], [yı, y2]} = xıy2 does not satisfy the conditions for an inner…

A: See the detailed solution below.

Q: Justify wether the following is an inner product or not =5x1y1-x1y2-x2y1+2x2y2 where x=(x1,x1) and…

A: By using axioms of inner product we have to show whether <x, y> is an inner product or not.

Q: Consider the polynomials a(x) = r' + x³ + x + 4 and b(x) = 2r³ + 4.x² + x + 2 from Z5[r]. a) Find…

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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: At least one of the answers above is NOT correct. Let f(z) = 2z¹ +22³ + 37² +2z + 1 and g(z) = 21³ +…

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Q: Using the inner product (,): P₂ (R) × P₂ (R) → R, (p, q) = p(0)q(0) + p(1)q(1) + p(2)q(2) calculate…

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Q: State and prove the : The characteristic polynomial of any diagonalizable linear operator T splits.

A: We have to prove that: The characteristic polynomial of any diagonalizable linear operator T splits.

Q: Let f(x)=-4, g(x) = -5x+7 and h(x) = -2x²-9x + 9. Consider the inner product (p,q) = p(-1)q(−1) +…

A: Step 1:Step 2: Step 3: Step 4:

Q: Show that (p(x), g(x)) = p(0)q(0) +p(1)q(1) + p(2)q(2) defines an inner product on P2.

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Q: 2d

A: Definition of P2​(R): Space of polynomials of degree at most 2 with real coefficients.Introduction…

Q: 7. Consider the inner product defined on P₂ by (p, q) = p(-1)q(−1) + p(0)q(0) + p(1)q(1). f₁(x) = x…

A: It is given that,  p, q = p(-1)q(-1) + p(0)q(0) + p(1)q(1) be an inner product on P2. f1(x) = x-x2 ,…