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Problem. Let F be a fixed n x n matrix and define a function T: Mnxn →Mnxn by T(X) = FX - XF. • Prove that I is a linear transformation. • The above item implies that the kernel of T is a subspace of Mnxni in English, describe the set of matrices in the kernel of T (hint: you will use the word "commute"). • If T' is the zero transformation (i.e. T(X) = 0 for all X), does that necessarily imply that F is the zero matrix? If it does, then prove it. If it doesn't, then give me an example of a nonzero matrix F where I is not the zero transformation.

Problem. Let F be a fixed n x n matrix and define a function T: Mnxn →Mnxn by T(X) = FX - XF. • Prove that I is a linear transformation. • The above item implies that the kernel of T is a subspace of Mnxni in English, describe the set of matrices in the kernel of T (hint: you will use the word "commute"). • If T' is the zero transformation (i.e. T(X) = 0 for all X), does that necessarily imply that F is the zero matrix? If it does, then prove it. If it doesn't, then give me an example of a nonzero matrix F where I is not the zero transformation.

Advanced Engineering Mathematics
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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Problem. Let F be a fixed n x n matrix and define a function T: Mnxn →Mnxn by T(X) = FX - XF. • Prove that I is a linear transformation. • The above item implies that the kernel of T is a subspace of Mnxni in English, describe the set of matrices in the kernel of T (hint: you will use the word "commute"). • If T' is the zero transformation (i.e. T(X) = 0 for all X), does that necessarily imply that F is the zero matrix? If it does, then prove it. If it doesn't, then give me an example of a nonzero matrix F where I is not the zero transformation.
Transcribed Image Text:Problem. Let F be a fixed n x n matrix and define a function T: Mnxn →Mnxn by T(X) = FX - XF. • Prove that I is a linear transformation. • The above item implies that the kernel of T is a subspace of Mnxni in English, describe the set of matrices in the kernel of T (hint: you will use the word "commute"). • If T' is the zero transformation (i.e. T(X) = 0 for all X), does that necessarily imply that F is the zero matrix? If it does, then prove it. If it doesn't, then give me an example of a nonzero matrix F where I is not the zero transformation.
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