Let V be the vector space of polynomials of degree < 2 with real coefficients, endowed with thestructure of an inner product space by settingf,8) := / 5Mg()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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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.
V= {[ xz +3 LX4 :-X₁ + 2x₂ + x₂ + 4xy=0 and X₁ + 3x₂ -X₂ - 2xy = 0 A Show that V is a subspace of Ru Be Find a basis for V and state dimension C. Replace your basis in part Bo with basis for V that consists of unit vectors
Find an orthonormal basis of the plane x12x2x3 = 0. 1 2 [廣告] ਦੇ Answer:
just b)
Find a basis for the plane 4x + 3y + 5z = 0 in R³. |
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'.
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 be the subspace of R3 with orthonormal basis {w1, w2}, where
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?
Could you explain how to solve this linear algebra question?
Consider the vector space V = R 3 over the field of real numbers F = R. Do the following vectors form a basis for R 3 ? , ,
(b) Let W = M2x1(C) be a vector space over C. Give a basis B for W.

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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 s.8):=/s (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.