Let x =Answer:-3]Nand y=-3Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R³ spanned by * and y.-58Hint: Find an orthogonal basis first, then normalize the vectors.

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Answered: Let + = Answer: !N W ][ and y=-5 F Use…

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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All vectors and subspaces are in R". Check the true statements below: OA. If {v1, V2, V3} is an orthogonal basis for W, then multiplying v3 by a scalar c gives a new orthogonal basis {v1, V2, CU3}. OB. If A = QR, where Q has orthonormal columns, then R= Q" A. C. The Gram-Schmidt process produces from a linearly independent set {r1,..., Tp} an orthogonal set {v1, ..., vp} with the property that for each k, the vectors v1, . same subspace as that spanned by x1,..., Pk. ·•, Uk span the
Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S {u1,..., uk} be an orthogonal basis for W and S' The goal of these exercises is to give two different arguments for the fact that {V1,..., Vi} be an othogonal basis for W-. (*) dim W + dim W- = n. 4. Show that SUS' is an orthogonal basis for R" and deduce (*).
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
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a b 6. Let H be a subspace of M2x2 whose vectors are of the form Then, B с 0 is a basis for H. 10 Find the coordinate vector of v = according to the basis, B.
(3) Let W be the subspace of R³ with orthogonal basis {u₁, u₂}, where 1 --0---0 = 1 -2 Let = -6 y = -3 1 2 Write y as the sum of a vector in W and a vector in W¹.
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.
a = -4 -3 -1 0 and b || -7 -3 11 -12 Use the Gram-Schmidt procedure to determine an orthonormal basis {₁, ₂} for the subspace of Rª spanned by the vectors a and b.

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Let + = Answer: !N W ][ and y=-5 F Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R³ spanned by and y. 8 ∞ J7 ₂ Hint: Find an orthogonal basis first, then normalize the vectors.