1. (-Let W₁-BE]>---CE)-and W₂ ==1 -1(a) Find a basis for W₁ W₂. What is dim(W₁ W₂)?(b) Show that R³ =W₁ + W₂.(c) Find a subspace U such that R³ = W₁ OU.be given subspaces of R³.

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Answered: 1. (- Let W₁ --E--E) and W₂ = = 1 -1…

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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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
2. The vector x = H [13] vector [2]B = [8] A 1 B 2 3 D0 in R2 relative to the orthogonal basis B What is the value of a + b? {0-[-]} has coordinate
4. (a) Find a basis for the subspace H of R spanned by the following vectors: 00000 1 (b) What is the dimension of H?
9 Find a basis and the dimension for the vector subspace W = ((x,y,z) e R³/ 2x-y+2=0 -8x+4y-4z=0} -6x+3y-3z0
(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.
10
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.
Let fi = 1+2x + 3x², f2 = 2 + 3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao +a1x+azx²[ao – 2a1 +a2 = 0} of R2[r]. Let g = 1+5x + 9x². If [g]g = (,) then s = a) -7 b) -5 c) 5 d) 7 e) g is not in V.
1) Let F, be the subspace generated by the vectors (0, 1, 1), (0,0.1) and F2 the subspace generated bythe vectors (1,1.0), (-1, 2, 0).is R° = F1 e F2
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+.
The vectors -6 01 = 02 = 03 =| 4 form a basis for R' if and only if k +

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1. (- Let W₁ --E--E) and W₂ = = 1 -1 (a) Find a basis for W₁ W₂. What is dim(W₁ W₂)? (b) Show that R³ =W₁ + W₂. (c) Find a subspace U such that R³ = W₁ OU. be given subspaces of R³.