5. Let V be the vector space of polynomials in t with inner product defined by(f, g) = [ f(t)g(t) dtApply the Gram-Schmidt algorith to the set {1, t, t2, t3} to obtain an orthonormalset {po, P1, P2, P3}

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Answered: 5. Let V be the vector space of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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' Define an inner product on the vector space P2(t) as following 3 (p(), g(t) -Σ α.)g (a,) i=0 where ao, a1, a2, a3 = -3, –1, 1,3 Construct the orthogonal bases from p(t) = 1, q(t) = t,r(t) = t? space P2(t).
7. Let V be the vector space of polynomials in two variables x and y of degree at most two: V = {ax² + bxy + cy² + dx + ey + ƒ | a, b, c, d, e, ƒ € R}. Let T be the linear operator on V defined by T(g(x, y)) = Ə Ix 9 (x, y) + Find the Jordan canonical form of T. Ə dy I (x, y). ду
5. Let V be a vector space, and let x1, X2, ... , Xk E V. 5(a) Prove that if either x1 = 0 or x;+1 E span{x1,...,x;} for some j = 1,2,... , k – 1, then x1, X2,.. dependent. %3D
7) Consider the vector space of odd real periodic function f:R Rwhere domain of these function are in the region-LSXSL, The base for odd periodic function sin (), Prove this basis is orthonormal. for any integer n
Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u, w) where u E U and wE W. Show that V is a vector space over K with addition in V and scalar multiplication on V defined by (4, w) + (u', w') = (u + d , w + w') and k(u,w) = (ku, kw) (This space V is called the external direct produd of U and W.)
Let V be a finite dimensional inner product space and T a non negative operator on V. Then T has a unique non negative square root, that is, there is one and only one non negative operator N on V such that N^2=T.
4. Let V = {ao+aj x+a2 x² | ao, a1, a2 E R} be a vector space of polynomial up to degree three, and p(x) = pPo+P1 x+p2 x², q(x) = qo+q1 x+q2 x² E V then show that = po.4o + pP1-q1 + P2-92 defines an inner product on V.
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4. Let the vector space P([-1,1]) have the inner product = f(t)g(t)dt, using Gram-Schmidt procedure, convert the basis {1, x,x²} into orthonormal basis.
2. An inner product on a vector space V is any mapping ,-) V x V R with the following properties: i. Positive definiteness: (x, x) 0 for all x E V and (r, x) 0 if and only if x 0 ii. Symmetry: (x, y) = (y, ) for all r, yE V ii. Linearity: (axby, z) a(x, z)b(y, z) for all r,y, z E V and a, b e R. Show the following a) The dot product y= iyi is an inner product on the vector space R". i-1 To (b) The product (x, y) = x(t)y(t) dt is an inner product on the vector space of functions
Consider the vector space R^4 and u = (u1, u2, u3, u4) ∈ R^4 . 1. Let Q : R^4 → R defined by Q(u) = |u3| + |u4| + sup{|u1|, |u2|}. Determine if Q is a norm, a seminorm or none of the two for R^4.

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5. Let V be the vector space of polynomials in t with inner product defined by (f, g) = [ f(t)g(t) dt Apply the Gram-Schmidt algorith to the set {1, t, t², t³} to obtain an orthonormal set {po, P1, P2, P3}