Definitions: Let T be a transformation from a vector space Vinto a vector space W • The kernel of T is the set {veV: T(v) = 0} • The range of T is the set {T(v): v=V}. • T is one-to-one if, for all u and v in V, T(u)=7(v) implies u = v. note: The kernel of T is a subspace of V and the range of T is a subspace of W. Let T be the transformation from M2×2 to M2x2 defined by T(A) = BA-AB where B = 1 A. Show that T is a linear transformation. B. Let A= a b cd . Express T(A) explicitly in terms of the parameters a, b, c, and d. C. Explain in your own words and without the use of mathematical symbols the meaning of kernel. 1 2 23 D. Is in the kernel of T? Justify your answer. E. Find a basis for the kernel of T. Show how you arrived at your basis. F. Show that Q = -1 -5 5 1 in the range of T by finding a specific matrix P such that T(P) =Q. Demonstrate that your matrix P satisfies T(P)=Q. G. Explain in your own words and without the use of mathematical symbols the meaning of range. H. Explain in your own words and with minimal use of mathematical symbols the meaning of one- to-one. Illustrate your explanation by drawing two simple graphs or diagrams showing the difference between a function that is one-to-one and a function which is not one-to-one. I. Is T one-to-one? Justify your answer.
Definitions: Let T be a transformation from a vector space Vinto a vector space W • The kernel of T is the set {veV: T(v) = 0} • The range of T is the set {T(v): v=V}. • T is one-to-one if, for all u and v in V, T(u)=7(v) implies u = v. note: The kernel of T is a subspace of V and the range of T is a subspace of W. Let T be the transformation from M2×2 to M2x2 defined by T(A) = BA-AB where B = 1 A. Show that T is a linear transformation. B. Let A= a b cd . Express T(A) explicitly in terms of the parameters a, b, c, and d. C. Explain in your own words and without the use of mathematical symbols the meaning of kernel. 1 2 23 D. Is in the kernel of T? Justify your answer. E. Find a basis for the kernel of T. Show how you arrived at your basis. F. Show that Q = -1 -5 5 1 in the range of T by finding a specific matrix P such that T(P) =Q. Demonstrate that your matrix P satisfies T(P)=Q. G. Explain in your own words and without the use of mathematical symbols the meaning of range. H. Explain in your own words and with minimal use of mathematical symbols the meaning of one- to-one. Illustrate your explanation by drawing two simple graphs or diagrams showing the difference between a function that is one-to-one and a function which is not one-to-one. I. Is T one-to-one? Justify your answer.
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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Can someone help me asnwer these questions thanks
Transcribed Image Text:Definitions: Let T be a transformation from a vector space Vinto a vector space W
•
The kernel of T is the set {veV: T(v) = 0}
•
The range of T is the set {T(v): v=V}.
•
T is one-to-one if, for all u and v in V, T(u)=7(v) implies u = v.
note: The kernel of T is a subspace of V and the range of T is a subspace of W.
Let T be the transformation from M2×2 to M2x2 defined by T(A) = BA-AB where B =
1
A. Show that T is a linear transformation.
B. Let A=
a b
cd
. Express T(A) explicitly in terms of the parameters a, b, c, and d.
C. Explain in your own words and without the use of mathematical symbols the meaning of kernel.
1 2
23
D. Is
in the kernel of T? Justify your answer.
E. Find a basis for the kernel of T. Show how you arrived at your basis.
F. Show that Q =
-1 -5
5 1
in the range of T by finding a specific matrix P such that T(P) =Q.
Demonstrate that your matrix P satisfies T(P)=Q.
G. Explain in your own words and without the use of mathematical symbols the meaning of range.
H. Explain in your own words and with minimal use of mathematical symbols the meaning of one-
to-one. Illustrate your explanation by drawing two simple graphs or diagrams showing the
difference between a function that is one-to-one and a function which is not one-to-one.
I.
Is T one-to-one? Justify your answer.
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