Suppose TE L(V) and By : v1, V2, ..., Vn is a basis for V such that for By 3 2 1] 0 1 0 0 0 2 M(T) (a) What is dim(V), ie what is n? (b) Without doing any calculations, what are the eigenvalues of T? Explain ho you know. (c) Suppose A is an eigenvalue for the matrix M(T) and u e F31 is an eigenvector associated with A, ie for a be F31. a M(T)u = Au = A b Prove that av1 + bv2 + cv3 is an eigenvector for T corresponding to eigenvalue A. (Hint: Recall Result 3.65 in the textbook which can also be written as: [Tu]By = M(T)[v)By for T E L(V). Also note that the linear map that takes v e V to [v]By E Fl is an isomorphism.)
Suppose TE L(V) and By : v1, V2, ..., Vn is a basis for V such that for By 3 2 1] 0 1 0 0 0 2 M(T) (a) What is dim(V), ie what is n? (b) Without doing any calculations, what are the eigenvalues of T? Explain ho you know. (c) Suppose A is an eigenvalue for the matrix M(T) and u e F31 is an eigenvector associated with A, ie for a be F31. a M(T)u = Au = A b Prove that av1 + bv2 + cv3 is an eigenvector for T corresponding to eigenvalue A. (Hint: Recall Result 3.65 in the textbook which can also be written as: [Tu]By = M(T)[v)By for T E L(V). Also note that the linear map that takes v e V to [v]By E Fl is an isomorphism.)
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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![Suppose TE L(V) and By : v1, V2, ..., Vn is a basis for V such that for By
3 2 1]
0 1 0
0 0 2
M(T)
(a) What is dim(V), ie what is n?
(b) Without doing any calculations, what are the eigenvalues of T? Explain how
you know.
(c) Suppose A is an eigenvalue for the matrix M(T) and u e F31 is an
eigenvector associated with A, ie for
a
be F31.
a
M(T)u = Au = A b
Prove that av1 + bv2 + cv3 is an eigenvector for T corresponding to
eigenvalue A. (Hint: Recall Result 3.65 in the textbook which can also be
written as: [Tu]By = M(T)[v]Bv for T E L(V). Also note that the linear
map that takes v e V to [v]By € Fl is an isomorphism.)
(d) Use the conclusion of part (c) to find the eigenspaces of T.
(e) Is T diagonalizable?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e038964-a669-472d-8392-8724d2c97ab7%2F8e89aec7-b233-4a7a-bc90-958beb8f5ce6%2Fo95ci39_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose TE L(V) and By : v1, V2, ..., Vn is a basis for V such that for By
3 2 1]
0 1 0
0 0 2
M(T)
(a) What is dim(V), ie what is n?
(b) Without doing any calculations, what are the eigenvalues of T? Explain how
you know.
(c) Suppose A is an eigenvalue for the matrix M(T) and u e F31 is an
eigenvector associated with A, ie for
a
be F31.
a
M(T)u = Au = A b
Prove that av1 + bv2 + cv3 is an eigenvector for T corresponding to
eigenvalue A. (Hint: Recall Result 3.65 in the textbook which can also be
written as: [Tu]By = M(T)[v]Bv for T E L(V). Also note that the linear
map that takes v e V to [v]By € Fl is an isomorphism.)
(d) Use the conclusion of part (c) to find the eigenspaces of T.
(e) Is T diagonalizable?
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