6. Let T : R³ → R³ be a linear transformation whose matrix is given by:Verify that: (1) the ordered setB =A =1-1[2 2₁[-2192 3-2 -346"-12-11is a basis of R³ and (2) each vector of B is an eigenvector of A. Lastly, find the matrixof T relative to the basis B, i.e., [T]B.

6. Let be a linear transformation whose matrix is given by:.Verify that:(1) the ordered set is a…

Answered: 6. Let T: R³ R³ be a linear…

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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1) Find the transition matrix P from the basis B={ (-1,2). (2, 1)} of R* to the basis {(4, 3). (-3,2). Ifliul, . find (u),
Let S P₂ P₂ be the linear transformation where S(x) = −12x − 30 and S(1) = 5x + 13. 0 Suppose Mg(S) = [-2] MB 0 3 Let Ɛ = {x, 1} be the standard basis of P2. Then P&B = 1 1 Then B = = {x+ x+ }
4. Let W be the xy-plane in R³. (a) Find an orthonormal basis ū₁, 2 for W. (b) Construct the matrix Q = [ ₁ ü₂ ] and calculate the matrix QQT. (c) Consider the linear transformation T: R³ → R³ that projects each vector orthogonally onto the xy-plane. Note that T() = QQT. Use the matrix to calculate T(₁), T(2), and T(ē3). (d) Explain how your answers make sense geometrically.
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9. Let and B 3 = {[B]-[-2]} = B' ={[2¹].[6]} be two ordered bases for R². Find the change of basis matrix [I]B'.
Suppose that T:R²→ R³ is a linear transformation and T(1,-3)=(4,–3, 1) and T(1,4)=(1,3, 1) and let B={(1,–3),(1,4)}. a) Find the matrix representation of Tin terms of B and S3 [T] where S3 is the standard basis for R'. b) Find the change of basis matrix [C, to convert from the standard basis in R? to B. c) Find [T] = [TCE, ->
Consider the following matrices: √/3 Q 8 M' = – 051010 100 10+ 9 (a) Find the component q in the coordinate transformation matrix Q such that the matrix is orthogonal and associated with rotation. q= = (b) Let Q transform the orthonormal basis set B to B'. Determine the components of matrix M' in the new orthonormal basis set B'. (c) Calculate the three invariants of M. I₁ I₂ = M = (d) Calculate the three invariants of M'. I I! = [2 −1 6 7 -5 6 1 -1 4 I3 = =
7. (a) Consider the ordered basis ß = [1,1 +x, 1 + x + x²] for V = P₂ and F = R. Suppose is in P2 and you know these coordinates: [] = (1, 2, 3). Find the vector 7. (b) Let A be a 2 x 2 matrix and a be a real number. Prove that det (a A)=a² det(A)

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6. Let T: R³ R³ be a linear transformation whose matrix is given by: → Verify that: (1) the ordered set A = B = 1 -1 2. 2 3 -2 -3 4 6 -{0·8·6) is a basis of R³ and (2) each vector of B is an eigenvector of A. Lastly, find the matrix of T relative to the basis B, i.e., [T]B.