5.Let (, ) be the Euclidean inner product on R", i.e. (x, y) = y"xfor r, y E R", and let || - || be the induced norm (so ||x||2 = (x, I)).(a) If A € M,n, show that (Ac, y) = {x, A™y).(b) Suppose P E Mnn satisfies P² - P and ||PP"x|| = || P"x|| for all a € R".%3DProve the following:i. |(PP" – p")x|| = 0 for all z E R".ii. P has a set of eigenvectors that form an orthonormal basis for R".

In the given question question given that inner product on Rn is defined as <x,y> = yTx and…

5. Let (, ) be the Euclidean inner product on R", i.e. (x, y) = y"x %3D for r, y E R", and let || - || be the induced norm (so ||¤||2 = (x, )). %3D (a) If A € Mn,n, show that (Ax, y) = (x, ATy). %3D (b) Suppose P E Mn.n satisfies P² = P and ||PP"I|| = ||P"r|| for all a € R". %3D Prove the following: i. |(PP" – p")x|| = 0 for all z E R". %3D

5. Let (, ) be the Euclidean inner product on R", i.e. (x, y) = y"x %3D for r, y E R", and let || - || be the induced norm (so ||¤||2 = (x, )). %3D (a) If A € Mn,n, show that (Ax, y) = (x, ATy). %3D (b) Suppose P E Mn.n satisfies P² = P and ||PP"I|| = ||P"r|| for all a € R". %3D Prove the following: i. |(PP" – p")x|| = 0 for all z E R". %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Similar questions

2. Find a norm on R2 for which ||(0, 1)|| = 1 = ||(1,0)|| and for which |(1, 1)|| = 0.000001. (Hint: try ||(a, b)|| = Ala + b| + Ba - b|.) %3D
9) Suppose u, v € V, where V is an inner product space, and ||u|| = ||v|| = 1 and (u, v) = 1. Find u – v.
I need help for this question
Q1/(A): Let A₁ = Z4 and A₂ = Z4 as Z -module. Find: 1- M = A₁ A₂. 3- All direct summand of M. 2- All Z -submodules of M. 4- Is M semisimple.
- Suppose P: V → V satisfies both P* = P and P2 = P. Prove that P is orthogonal projection onto Ran(P). You can use the fact (that we will learn soon) that (Ax, y) = (x, A*y) for all x, y = V.
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6. 'The mapping T :R³ → R² is defined as T (x, y, z) = (x + 2y+ 3z, x + 3y + 2z) is a linear transformation from R³ into R². Then %3D its Kernel T is (a) L {c} where a = (- 1, 5, 1) (b) L{a} where a = (- 1, 1, 5) (ç) L{a} where a = (- 5, 1, 1) (d) None of these
7. Consider the set of vectors S = {u = (-1, 2, – 1), v = (2, – 5, 0), w = (1, – 3, – 1)}, then find the vector orthonormal projection of "u" on "v" and the vector orthonormal projection of "u" on "w" respectively.
11. If T and H are invertible linear operators on a vector space V(F), then show that 1 TH is invertible and (TH)'=H'T'. %3D
Let a1 = col [1, 1,0] and r2 = col [1, 1, 1]. (a) Apply GSOP to (1,x2) to generate an orthonormal set ( v1, v2 ). %3D %3D (b) Find orthogonal projections of x span (1,22 ). col [0, 1, 1] on S1 = span (r1 ) and on S2