Jump to level 1WA[4]34+0-19-81125117find an orthogonal basis under the Euclidean inner product.LetOrthogonal basis:a = Ex: 1.23==[4]2b Ex: 1.233-14]10-13-8V256 V3173be a basis for R³. Use the Gram-Schmidt process toc Ex: 1.23abA

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Answered: Jump to level 1 -34 51 17 find an…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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1 2 -1 1 (VI). (12') Let A = 0 2 -1 Find a basis for the row space of A and a 1 4 -2 1 basis for N(A).
Let W = Span u = U2 = U3 = When applying the Gram-Schmidt process to the vectors U,u2,u3 (in this order) to find an grthonormal basis V1,V2,V3, what is V3? 1 1 1
18. Use the Gram-Schmidt process to transform the given basis {w1, w2, W3} into an orthogonal basis where wi = (1,1,2), w2 = (0,1,1), and w3 = (-1,2,1).
15. Assume that the vector space R³ has the Euclidean inner product. Apply the Gram-Schmidt process to transform the basis vectorsu₁ = (1,1,1), U₂ = (-1,1,0), u3 = (1,2,1) into an orthogonal basis {V₁, V₂, V3} and then normalize the orthogonal basis vectors to obtain an orthonormal basis {91,92,93}.
The standard basis & (e₁,e₂) and two custom bases B (b₁,b₂} and C (c₁, c₂} for R² are shown in the figures below. Standard basis S= (₁, ₂) 4 a. Find the change of basis matrix from custom B-coordinates to standard S-coordinates. [id] = id= -2 b Find the change of basis matrix from custom C-coordinates to standard S-coordinates [id] = -1 -1 -1 c. Find the change of basis matrix from B-coordinates to C-coordinates -3 -1/12 1/4 d. Find the change of basis matrix from C-coordinates to B-coordinates. 3 2 1 -1 -2 -9 b2 -3-2-1 82 el bl 1 2 3 b1 X Custom basis B = {b₁,b₂) |id|s [id Standard basis S= (0₁,0₂} a 3 2 1 -1 -2 -9 c2 -3 -2 -1 G2 62 217 el [id] ↑ cl 12 2 3 61 H x Custom basis C (C₁, C₂)
The set -23 (-0-0-0-0) 16 is a basis for R Use the Gram-Schmict process to produce an orthogonal basis for ¹. 111 = V₁ = U₂ = V₂ 11 VJ -15 -3 your answer" 11₁ = V₁ U₂ V₂. 11111 To make computations nice, if necessary, we can scale this vector by a non-zero scalar, so that y= ū₁ = us -U₁= 11V₂ HH 11₁ all To make computations nicer, if necessary, we can scale this vector by a non-zero scalar, so that u₂ = ₂ Va 1₂ U₂ u₂ V4. u₁u₂ 1₁ (Do not scale your answer -= -u₂ = 11₂ 11 (Do not scale r (Do not scale your answer) To make computations nicer, if necessary, we can scale this vector by a non-zero scalar, so that Therefore, [u, uz, us. u) is en orthogonal basis for R. To obtain an orthorormal basis, we scale each of these vectors as follows: Û₂ = u;= 3 1 113 VI 113 113 -1₂ = |u₁|| -1₂ = Note: To enter a number of the form type a/sqrt(n) Thus, (₁, ₂, 3, 4) is an orthonormal. basis for R 113= -6 -6 -15
Let + {U₁ = [₁ ], U₂₁ = [2₂8] · U₂ = [-2² , U3 20 Gram-Schmidt process to find an orthogonal basis under the Frobenius inner product. Orthogonal basis: V₁ a = Ex: 5 =[]} a 2 {v₁ = [1₂8], v₂ = [8²], b = Ex: 5 3], V₁ = [1 V3 c = Ex: 1.23 -1.33 1.33 be a basis for a subspace of R2x2. Use the d= a]} Ex: 1.23
1 3. (а) Find an orthonormal basis B {ū1, ū2, đ3} of R³ such that u and the x-component of ūz is 1/3. [1 Find the coordinate vector of 3 relative to the basis B. (b) 1
show that a set of vectors{et cos(2t), e¯t sin(2t)} form a basis of solution space for x" + 2x' + 5x = 0
21 U2 = -2] Let uj = u2 1 and y = 2 [3. and W= Span{u1,U2}. (a) Find an orthogonal basis for W. (b) Find the closest point in W to y. (c) What is the shortest distance between y and W?

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Jump to level 1 -34 51 17 find an orthogonal basis under the Euclidean inner product. Let Orthogonal basis: a= Ex: 1.23 U₂ = V1 = b= Ex: 1.23 2 -[]} , Ug = V₂ = -14 56 17 V3 c = Ex: 1.23 be a basis for R³. Use the Gram-Schmidt process to a E