Consider the vector space R of sequences with an infinite tail of zeroes. In other words, given a sequence ER one can find a number N such that all entries after the N-th entry are zeroes. Now consider two linear transformations S and T defined as follows: Let S(a₁, A2, A3, ...) = (a₂, A3, ...). In other words, S moves the sequence to the left forgetting the first element. Let T(A₁, A2, A3, ...) = (0, A₁, A₂, A3, ...). In other words, T moves the sequence to the right adding a 0 in the beginning of the sequence. It is possible to show that S and T are linear transformations? Consider the composition of these transformations. Recall that (S. T)(v) = S(T(v)) and (T · S)(v) = T(S(v)). (a) Is λ = 1 an eigenvalue of S. T? If so, describe the eigenvectors in its corresponding eigenspace E₁. If not, show that no vector v ER satisfies the definition of eigenvector for eigenvalue 1. ∞o (b) Does ST have any other real eigenvalues? If so, describe the set of all real eigenvalues of S. T. For an arbitrary eigenvalue of S. T, describe its eigenvectors. If not, show that no vector v ER satisfies the definition of eigenvector for eigenvalue where is an arbitrary element of R \ {1}. (c) Is λ = 1 an eigenvalue of TS? If so, describe the eigenvectors in its corresponding eigenspace E₁. If not, show that no vector v ER satisfies the definition of eigenvector for eigenvalue 1. Note: you will not be able to switch to the language of matrices, as R is not finite-dimensional.
Consider the vector space R of sequences with an infinite tail of zeroes. In other words, given a sequence ER one can find a number N such that all entries after the N-th entry are zeroes. Now consider two linear transformations S and T defined as follows: Let S(a₁, A2, A3, ...) = (a₂, A3, ...). In other words, S moves the sequence to the left forgetting the first element. Let T(A₁, A2, A3, ...) = (0, A₁, A₂, A3, ...). In other words, T moves the sequence to the right adding a 0 in the beginning of the sequence. It is possible to show that S and T are linear transformations? Consider the composition of these transformations. Recall that (S. T)(v) = S(T(v)) and (T · S)(v) = T(S(v)). (a) Is λ = 1 an eigenvalue of S. T? If so, describe the eigenvectors in its corresponding eigenspace E₁. If not, show that no vector v ER satisfies the definition of eigenvector for eigenvalue 1. ∞o (b) Does ST have any other real eigenvalues? If so, describe the set of all real eigenvalues of S. T. For an arbitrary eigenvalue of S. T, describe its eigenvectors. If not, show that no vector v ER satisfies the definition of eigenvector for eigenvalue where is an arbitrary element of R \ {1}. (c) Is λ = 1 an eigenvalue of TS? If so, describe the eigenvectors in its corresponding eigenspace E₁. If not, show that no vector v ER satisfies the definition of eigenvector for eigenvalue 1. Note: you will not be able to switch to the language of matrices, as R is not finite-dimensional.
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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Transcribed Image Text:Consider the vector space R of sequences with an infinite tail of zeroes. In other words, given a sequence
ER one can find a number N such that all entries after the N-th entry are zeroes.
Now consider two linear transformations S and T defined as follows:
Let S(a₁, A2, A3, ...) = (A₂, A3, ...). In other words, S moves the sequence to the left forgetting the first
element.
Let T(a₁, A2, A3, ...) = (0, A₁, A₂, A3, ...). In other words, T moves the sequence to the right adding a 0 in the
beginning of the sequence.
It is possible to show that S and T are linear transformations? Consider the composition of these
transformations.
Recall that (S. T)(v) = S(T(v)) and (TS)(v) = T(S(v)).
(a) Is λ = 1 an eigenvalue of S. T? If so, describe the eigenvectors in its corresponding eigenspace E₁. If not,
show that no vector v ER satisfies the definition of eigenvector for eigenvalue 1.
(b) Does S. T have any other real eigenvalues? If so, describe the set of all real eigenvalues of S. T. For an
arbitrary eigenvalue of S. T, describe its eigenvectors. If not, show that no vector v ER satisfies the
definition of eigenvector for eigenvalue where is an arbitrary element of R \ {1}.
If not,
(c) Is λ = 1 an eigenvalue of T. S? If so, describe the eigenvectors in its corresponding eigenspace E₁.
show that no vector v ER satisfies the definition of eigenvector for eigenvalue 1.
Note: you will not be able to switch to the language of matrices, as R is not finite-dimensional.
∞
∞
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