Exercise 4.7 Consider, for varying a € R, the matrix:α0110 222 a +1Aa =a) Establish if the subspace ((1,0,0), (0, 1,0)) contains eigenvectors for Aa.b) Determine the eigenspaces of Aa.c) Say for which a ER the matrix is diagonalisable.

Consider, for varying α∈ℝ, the matrix: Aα=01α10222α+1 a) Establish if the subspace 1,0,0,0,1,0…

Answered: Exercise 4.7 Consider, for varying a €…

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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Let a € R, and define vectors u₁, U₂, U3 € R³ in terms of a by - 8. a U₁ = 2 U₂ = 3a + 3 2a +3 a² + 3a 7 [4a - 2] 13 = 3a-1 2a² (a) Show that the rank of the matrix [u₁ u₂ u3] does not depend on a. Hint: Find the rank by putting the matrix in row-echelon form. (b) Making reference to a fact from the course, show that u₁, U₂, U3 never span R³, no matter what value a takes. (c) Express u3 as a linear combination of u₁ and u2, i.e., as c₁u₁ + c₂U2. The scalars c₁ and c₂ will depend on a.
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Exercise 4.7 Consider, for varying a € R, the matrix: 01 a 0 2 2 2 a +1 Aa = a) Establish if the subspace ((1,0,0), (0, 1,0)) contains eigenvectors for Aa. b) Determine the eigenspaces of Aa. c) Say for which a € R the matrix is diagonalisable.