Problem 8: Suppose €1, €2,vectors in V such thaten is an orthonormal basis of V and V₁, V2.... 91||ej - vj|| <for each j. Prove that V₁, V2...., Un is a basis of V.9-Un are(Hint: Suppose there existed a non-trivial linear relation av0. Considerw = 1 ajej. Then ||w|| = ||w-0|| = || Σ;=1 aj(e;-v₁)||. Now try using the triangleand Cauchy-Schwarz inequalities to the last sum and arrive at a contradiction.)

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Answered: Problem 8: Suppose e₁,e2, ..., en is an…

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 the basis B = if the coordinates of & relative to basis B are [w] = B 13 || 2 3 (100) -3 5 2 4 LL 1 }] 5 find w
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If T is defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A = 1 -7 -25 -3 12 39 and b = -4 3 Find a single vector x whose image under T is b. 3 -A X = Is the vector x found in the previous step unique? O A. No, because there are no free variables in the system of equations. O B. No, because there is a free variable in the system of equations. O C. Yes, because there is a free variable in the system of equations. O D. Yes, because there are no free variables in the system of equations.
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Suppose now we have a set of non standard basis vectors v1, v2, .., Un e R" b) which satisfies the following equation: 1 1 WV = I = [v1, v2, ..., vn] where w and vi are the rows and columns of W and V, respectively. Any vector z € R" can be written as a linear combination of the st andard basis vectors v1, v2, ..., Vn as follows: k=1 Determine the coefficients ak in terms of z and wf only.
1. Three vectors are given by their coordinates: u =| w = Solve the necessary problem(s) and select exactly one right answer from (a), (b, (c), (d): (a) {u,v, w} is a basis in R (b) {u,v, w} do not span R (c) {u,v,w} is a linearly independent set (d) {u,v} is a basis in R² , but {v, w} is not a basis in R² . 2. The following vectors a, b, c, and v are given by their coordinates: [1] = 1 [-27 [1] a = 0, b= 1 |2] 3 Solve the problem(s) that will allow you to answer 6 the following questions, and then select exactly one right answer from (a), (b, (c), (d): Is the set {a,b,c} a basis in R ? Does the set {a,b, c} span R ? Does the set {a,b,c} span vector v? (a) {a,b,c} is not a basis in R³, and it does not span R , but {a,b,c} does span vector v (b) {a,b,c} is a basis in R³ , so it spans R'and it spans v as a vector from R. (c) {a,b,c} is not a basis in R , so it does not span R , and it does not span vector v. (d) None of (a), (b), (c) is true.
subject Linear algebra
et △ABC be a triangle in S^2 (the two-dimensional sphere). Let the dual point C'∈S^2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, andit follows (you may assume) that △A' B' C' is too.(question) Are there triangles △ABC in S^2 identical to their own dual △A'B'C'? I would be very thankful if you could provide some explanation with the steps, thank you in advance.
Given the set S = {M1, M2, M3, M4}, where 3 1 M1 M2 2 4 1 -3 0 -2 1 1 M3 M4 4 1 2 -1 (a) Show that S is a basis of M22. (b) Find the coordinates of 20 M = 8 29 34 relative to S. 2.

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