1) We write vectors horizontally for this problem. Let U be the subspace of RS definedbyU = {(x1, X2, X3, X4, x3)|X1, X2, X3, X4, X5 E R, x, = 3x, and x3 = 7x4}.a) Find a basis of Ub) Extend the basis in part (a) to a basis of RSc) Find a subspace W of R5 such that R5 = U OW.

Answered: Image /qna-images/answer/b7721764-ec21-4877-b9ee-75c634ad1787.jpg

Answered: 1) We write vectors horizontally for…

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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Given basis B = {(1, -1), (2, 3)} and basis C = {(1, 0), (0, 1)}, and linear map T = (x + 2y ; 2x - y) from R2 --> R2, find the coordinate vectors of basis C in terms of basis B.
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
Consider the basis B = {({2, 1), (3, 1}} for V = R2 and let {x1, x2} be the dual basis of B. Then for an arbitrary pair (c, d) EV, the set {x1(c, d), x2(c,d)) is equal to: {(3d-c), (c-2d)} {(3d+ c), (c+ 2d)} {(2d-c), (c + 3d)} {(2d+c), (c + 3d)} {(3d+c), (2c - d)}
just 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).
Consider the subspace S of the Euclidean inner product space R+ spanned by the vectors v₁ = (1,1,1,1), v₂=(1,1,2,4), v₂=(1,2,-4,-3). Find an orthogonal basis of S. O*((-.-3.1).(1.2.-4,-3).(4,2,1,1)) OB. © ³ {(1,1,2,4), ( − 1, − 1,0,2 A. -1,0,2), (12. 2.-3,1)} O C. None in the given list. OD. {(1,1,1,1),(1,1,2,4), (1,2,-4,-3) } OE {(1,1,1,1),(-1,-1,0,2), (1,3, -6,2) }
Consider the basis B = {({2, 1), (3, 1}} for V = R2 and let {x1, x2} be the dual basis of B. Then for an arbitrary pair (c, d) EV, the set {x1(c, d), x2(c,d)) is equal to: {(3d-c), (c-2d)} {(3d+ c), (c+ 2d)} {(2d-c), (c + 3d)} {(2d+c), (c + 3d)} {(3d+c), (2c - d)}
Could you explain how to solve this linear algebra question?
a) Find , \pl, d (p, q), and then find the angle between them for the polynomials in P2 if p(x) = -3x² + 2x – 5 , and q(x) = -x+5x? b) Verify that the basis set S = {(3,0,4), (-4,0,3), (0,1,0)} is orthogonal then find the coordinates of the vector u = (-5,5,1) using inner product
(3,0,0), v2 = (1,2,0), and v3 = a) Are the vectors linearly independent or linearly dependent? Given vectors v, = (2, –4,5). b) Do the vectors form a basis for R3? Why or why not?

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1) We write vectors horizontally for this problem. Let U be the subspace of IRS defined by U = {(x1,x2, X3, X4, Xs)|X1, X2, X3, X4, X5 € R, x1 = 3x2 and x3 = 7x4}. a) Find a basis of U b) Extend the basis in part (a) to a basis of R c) Find a subspace W of R$ such that RS = U OW.