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{ X11 X12 M2 : Xij E R X 22 X21 be the vector space of 2 × 2 real matrices with entrywise addition and scalar multiplication. Let [1 A 0 2 and consider the linear transformation T : M2 → M2 defined by T(X) — АХ —ХА. (a) Find a basis of Ker(T). what is the dimension of Ker(T)? (b) Find an eigenvector of T with strictly negative eigenvalue, (i.e., a nonzero X E M2 such that T(X) = XX for somel< 0) or show that no such eigenvector exists.

{ X11 X12 M2 : Xij E R X 22 X21 be the vector space of 2 × 2 real matrices with entrywise addition and scalar multiplication. Let [1 A 0 2 and consider the linear transformation T : M2 → M2 defined by T(X) — АХ —ХА. (a) Find a basis of Ker(T). what is the dimension of Ker(T)? (b) Find an eigenvector of T with strictly negative eigenvalue, (i.e., a nonzero X E M2 such that T(X) = XX for somel< 0) or show that no such eigenvector exists.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Let { x11 X 12 M2 : xij E R X21 X22 be the vector space of 2 × 2 real matrices with entrywise addition and scalar multiplication. Let [1 A = and consider the linear transformation T' : M2 → M2 defined by T(X)= AX ХА. (a) Find a basis of Ker(T). what is the dimension of Ker(T)? (b) Find an eigenvector of T' with strictly negative eigenvalue, (i.e., a nonzero X E M2 such that T(X) = \X for some < 0) or show that no such eigenvector exists.
Transcribed Image Text:Let { x11 X 12 M2 : xij E R X21 X22 be the vector space of 2 × 2 real matrices with entrywise addition and scalar multiplication. Let [1 A = and consider the linear transformation T' : M2 → M2 defined by T(X)= AX ХА. (a) Find a basis of Ker(T). what is the dimension of Ker(T)? (b) Find an eigenvector of T' with strictly negative eigenvalue, (i.e., a nonzero X E M2 such that T(X) = \X for some < 0) or show that no such eigenvector exists.
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