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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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?
The coordinates for the vector v = (1,2,4) relative to the standard basis B = {(1,0,0), (0, 1,0), (0,0,1)} is [v] B = (1,2,4). The coordinates for v relative to the nonstandard basis B' = {(1, 0, -1), (2,1,-1), (-2, 1,4)} is [v]B¹ = (9,−1,3). Find the transition matrix P-1 from B to B' and show that this does change the coordinate of v relative to B to the coordinate of v relative to B'.
Please
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?
Let V₁ = ,V₂ = 1 and let W be the subspace of R¹ spanned by V₁ and v2. (a) Convert {V1, V2} into an orhonormal basis of W NOTE: If your answer involves square roots, leave them unevaluated. Basis = {}. (b) Find the projection of b = 1 onto W (c) Find two linearly independent vectors in R¹ perpendicular to W. Vectors = { Note: You can earn partial credit on this problem.
V = Span ({··]}) Find an orthonormal basis of R³ that contains the vector 圓

4. Consider the subspace V of R' given by 1 3 V = span (a) Find an orthonormal basis for the orthogonal complement V-. (b) Find the closest point to (1, 1, 1, 1) in V+.
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.