4. Let V be a vector space over C and let L : V → V be a linear transformation. (a) State the definition of an eigenvector of L. (b) Let A e C be arbitrary. State the definition of V,, the A-eigenspace of L, and prove that it is a subspace of V. (c) Find the characteristic polynomial Pa(t) of the matrix A given below, and then the eigen- values of A. -1 -1 2 A = 2 2 1 -1
4. Let V be a vector space over C and let L : V → V be a linear transformation. (a) State the definition of an eigenvector of L. (b) Let A e C be arbitrary. State the definition of V,, the A-eigenspace of L, and prove that it is a subspace of V. (c) Find the characteristic polynomial Pa(t) of the matrix A given below, and then the eigen- values of A. -1 -1 2 A = 2 2 1 -1
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:4. Let V be a vector space over C and let L : V → V be a linear transformation.
(a) State the definition of an eigenvector of L.
(b) Let A e C be arbitrary. State the definition of V,, the A-eigenspace of L, and prove that it
is a subspace of V.
(c) Find the characteristic polynomial Pa(t) of the matrix A given below, and then the eigen-
values of A.
-1 -1
2
A =
2
2
1.
-1
(d) Choose one of the eigenvalues of A you found in (c), and find an eigenvector of A corre-
sponding to this eigenvalue.
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