Question 5.Let α(e1, e2, e3) and B = (e2, e3, e1). Here e; is the ith standard basis vector of R³.Let T : R³ → R³ be given by T(x, y, z) = (x + y, Y +x, 2z). (You may assume this map is linear.)Compute MBa (T) and P3-a-

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Answered: Question 5. let α- (e1, e2, e3) and ß =…

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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6. Is the set S {(2,3, 1), (1,2, 1), (-2, 3, 3)} C R linearly %3D independent or dependent? Justify your answer. 7. Prove that the set S = {(1,3, 2), (3, 1, 2), (0, 1,0)} spans R. 8. Find a basis for R that includes the vectors (1,0, 2) and (0, 1, 1). Justify your answer. 9. Show that the function T: R → R defined by T(x, y, z) = (x – z, 2x + y, 2y – 3r) is a lincar transformation. 10. Define the function T: R3 → R as 2 1 T(v) = Av = -3 2 0 1 0- 0-1 Find the image of T(-4,5,1) and the preimage of (-1,2, 2). 11. Find the eigenvalues and the corresponding eigenvectors of the 2 matrix A 3
7. Consider the set of vectors S = {u = (-1, 2, – 1), v = (2, – 5, 0), w = (1, – 3, – 1)}, then find the vector orthonormal projection of "u" on "v" and the vector orthonormal projection of "u" on "w" respectively.
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5. Find the standard matrix A and A' for T = T2° T1 and T' = T1° T2, where T1:R2 → R3, T1(x,y) = (x,x+y,y) and T2:R3 → R², T2(x,y,z) = (0,y). Use standard basis vectors to derive your re- sults.
Y, B d X A C d = 8, a = T/3 rad. Suppose T is a transformation matrix, where cos 0 - sin 0 T = [sin 0 cos 0 (1) State the vector columns of all triangle vertex. (2) Form a matrix, named A, that is collection of all the vertex.
43. Linear combination Let u = (1, 2, 1), v = (1,-1,-1), w = (1, 1,-1), and z = (2, -3,-4). Find scalars a, b, and c such that z = au + bv + cw.
8. Let A = | 1 -3 -5 -7 7 5 (a) Find the image of u = and define T: R³ R2 by T(x) = Ax. 2 1 under T. (b) Find a vector x whose image under T is b = -12 12 Explain why your work makes sense.
-3 Consider the matrix A = and | () the vector u = Let B = (A – A" )² and define vector V = Bu. If y = U2, write down the value of v .
2
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1. Find the length of a vector || u +v || u= ( 1,2,1) V= v3D(0,2,-2)
3. Find the vector component of u along a and the vector component of u orthogonal to a for the given vectors u = (2,0,1) and a = (1,2,3)

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Question 5. let α- (e1, e2, e3) and ß = (e2, e3, e1). Here e; is the ith standard basis vector of R³. Let T : R³ → R³ be given by T(x, Y, z) = (x + y, Y +x, 2z). (You may assume this map is linear.) Compute MBa(T) and PBe-a-