DO NOT WANT AI SOLUTIONS. Thank you 1. Consider the mapping (,): R2 x R² → R defined by(u,v) = u³ Du, Vu, v € R²,where=»-(62)such that d1, d2 >0. Prove that (,) is an inner product.

Here's a proof that the given mapping defines an inner product on ℝ²: 1. Symmetry:To prove that we…

Answered: 1. Consider the mapping (,): R2 x R² →…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 12EQ

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Let I = ff, (a2 – 2) da dy, where D = {(*, 3) : 1 0, y 2 0} Show that the mapping u = sy, v = * – y maps D to the rectangle R = [1, 4] x (0, 6]. (a) Compute 8 (*, v)/8(u,v) by first computing 8 (u, v)/a(*, 3). = v over R and evaluate. (b) Use the Change of Variables Formula to show that I is equal to the integral of f (u, v): (a) a(4) = (b)I :
Let q = - 2xy - y² + 2xz+2yz + z² be a quadratic form on R³ viewed as a polynomial in 3 variables. Find a linear change of variables tou, v, w that puts q into the canonical form in `Sylvester's law of inertia'!. What are the values of the associated indices s, t? Select one: O We let u = √3(x − y + 2), v = x+y, w = (x - y)/2 to find that q = u²+². Hence s = 2, t = 0 in Sylvester's law of intertia. O We let u = x - 2y, v = z+y, w = √3y to find that q=u²v² w². Hence s = 1,t = 2 in Sylvester's law of intertia. O we let u = x, v= √2(x+y), w = x+y+z to find that q = u²v² + w²2. Hence s = 2, t = 1 in Sylvester's law of intertia. O None of the others apply O The quadratic form does not obey the condition to be diagonalisable over R. This is because the minimal polynomial of the corresponding matrix is not a product of distinct linear factors. Hence Sylvester's law of inertia does not apply. By convention, we set s = t = ∞ when this happens.
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1. Consider the mapping (,): R2 x R² → R defined by (u,v) = u³ Du, Vu, v € R², where = »-(62) such that d1, d2 >0. Prove that (,) is an inner product.