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A. Show that T is a linear transformation. a b В. Let A%3 . Express T(A) explicitly in terms of the parameters a, b, c, and d. с d C. Explain in your own words and without the use of mathematical symbols the meaning of kernel.

A. Show that T is a linear transformation. a b В. Let A%3 . Express T(A) explicitly in terms of the parameters a, b, c, and d. с d C. Explain in your own words and without the use of mathematical symbols the meaning of kernel.

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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Definitions: Let T be a transformation from a vector space Vinto a vector space W The kernel of Tis the set {veV:T(v) = 0} The range of Tis the set {T(v): veV}. Tis one-to-one if, for all u and v in V, T(u) =T(v) implies u = v. note: The kernel of T is a subspace of V and the range of T is a subspace of W. 11 Let T be the transformation from M2x2 to Mx2 defined by T(A) = BA – AB where B 11 A. Show that Tis a linear transformation. a b . Express T(A) explicitly in terms of the parameters a, b, c, and d. c d B. Let A= C. Explain in your own words and without the use of mathematical symbols the meaning of kernel. 1 2 D. Is in the kernel of T? Justify your answer. 2 3 E. Find a basis for the kernel of T. Show how you arrived at your basis. -1 -5 F. Show that Q in the range of I by finding a specific matrix P such that T(P) =Q. 5 1 Demonstrate that your matrix P satisfies T(P) =Q.
Transcribed Image Text:Definitions: Let T be a transformation from a vector space Vinto a vector space W The kernel of Tis the set {veV:T(v) = 0} The range of Tis the set {T(v): veV}. Tis one-to-one if, for all u and v in V, T(u) =T(v) implies u = v. note: The kernel of T is a subspace of V and the range of T is a subspace of W. 11 Let T be the transformation from M2x2 to Mx2 defined by T(A) = BA – AB where B 11 A. Show that Tis a linear transformation. a b . Express T(A) explicitly in terms of the parameters a, b, c, and d. c d B. Let A= C. Explain in your own words and without the use of mathematical symbols the meaning of kernel. 1 2 D. Is in the kernel of T? Justify your answer. 2 3 E. Find a basis for the kernel of T. Show how you arrived at your basis. -1 -5 F. Show that Q in the range of I by finding a specific matrix P such that T(P) =Q. 5 1 Demonstrate that your matrix P satisfies T(P) =Q.
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