Let V be the vector space of polynomials of degree < 2 with real coefficients, endowed with thestructure of an inner product space by setting(f,g):=f(t)g(t)dt.Produce an orthonormal basis for V by applying the Gramm-Schmidt orthogonalisation process to thebasis (1, x, x²) of V.

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Answered: 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

Section: Chapter Questions

Problem 1RQ

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Consider the Euclidean inner product space R³ with a basis B={(1,1,1),(0,1,1),(0,0,1)}. Find an orthonormal basis of R³. 0^{(1,2,1),(0,-1,-1),(0,1,0)) 08((1.1.1).(-2,1,1),(0,1,1)) 0(-2,1,1),(0,1,1),(,-1,1)} O A. O B. OD. None in the given list O E. {(1,1,1),(0,0,1),(0,-1,1)}
just b)
The set of vectors {(1, 1, 0), (1, 0, 1), (0, 0, 1)} is an orthonormal basis for the space R^3 in some scalar product <.,.>. Then, the scalar product <(1, 1, 1), (1, 0, 0)> is equal to: (a) -1. (b) 0. (c) 1. (d) 2. (e) 4. The correct answer is (a). Why?
Answer the following questions about the Gram-Schmidt process for obtain- ing an orthogonal basis {v1, V2, V3} for the span of 2 5 1 {u,, u2, U3} -3 1 (a) what is ||v ||? (b) what is u2 · Vị (c) what is v2 (d) what is ||V2||? (e) what is uz · V1 (f) what is uz · V2 (g) what is v3 (h) what is v3 · V2
Use the Gram-Schmidtorthonormalization process to find an orthonormal basis of ?2(with the dot product)from the basisthe basis{(4,−3,0),(1,2,0),(0,0,4)}.
4 Let W = span(vV1, V2) where vi = and v2 2 . Then: (a) Find the orthogonal decomposition of the vector x = 3 (b) Find an orthogonal basis for W- (remember to justify your answer).
Let w,=(1,0,0,1) and w=(-1,0,0,1). Let W=(w | w w, = w-w, = 0}, where w-w, denotes the dot product of w, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique?
Do the following. Considering the subspace W of R^4 , which has as its basis the set B = 2 (a) Explicitly describe the orthogonal complement W+ of W and give a basis and the dimension of this subspace; (b) Consider the following FACT from theory (theorem): If v1,v2, .. , vr is an orthogonal basis of a subspace W of R" and v €R" then the vector projection projw (v) of v onto W is given by : projw (v) = proju, (v) + projuz(v) +...+ projo, (v). It is further proven that this is the vector of W closest to the vector v. Given the result above, find the vector of W closest to 1 E R'. v = 3

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Let V be the vector space of polynomials of degree < 2 with real coefficients, endowed with the structure of an inner product space by setting (f,g):= f(t)g(t)dt. Produce an orthonormal basis for V by applying the Gramm-Schmidt orthogonalisation process to the basis (1, x, x²) of V.