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Let ēj = (1,0,0)T, ē2 = (0, 1,0)", ēz = (0,0, 1)T be the standard basis of R³. Consider a linear map T : R³ → R³ satisfying T (x, Y, z)“) = (0,0,0)" whenever 2x – y + z = 0. (a) -Shew that T(,0, H7)-(0,0,)T and T((0,1,1)7)-(0,0,0)"- (b) Justify whether T is an isomephism using resut from (a).– (c) Given T(ē3) = (1, 2, 3)". Find T (ē1) and T (ē2) by using result from (a).

Let ēj = (1,0,0)T, ē2 = (0, 1,0)", ēz = (0,0, 1)T be the standard basis of R³. Consider a linear map T : R³ → R³ satisfying T (x, Y, z)“) = (0,0,0)" whenever 2x – y + z = 0. (a) -Shew that T(,0, H7)-(0,0,)T and T((0,1,1)7)-(0,0,0)"- (b) Justify whether T is an isomephism using resut from (a).– (c) Given T(ē3) = (1, 2, 3)". Find T (ē1) and T (ē2) by using result from (a).

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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Standard basis

Let ēj = (1,0,0)", ē2 = (0, 1,0)", ēz = (0,0, 1)" be the standard basis of R3. Consider a linear map T : R³ → R³ satisfying T ((x, y, z)²) = (0, 0, 0)" whenever 2x – y + z = 0. (a)-Shew that T(o, 1)7)–(0,0,0)" and T ((0,1,17)-(0,0,0)7. (b) Jstify whether T is en isomerphiem using result from (a) | (c) Given T(ē3) = (1,2, 3)". Find T (ei) and T (ē2) by using result from (a).
Transcribed Image Text:Let ēj = (1,0,0)", ē2 = (0, 1,0)", ēz = (0,0, 1)" be the standard basis of R3. Consider a linear map T : R³ → R³ satisfying T ((x, y, z)²) = (0, 0, 0)" whenever 2x – y + z = 0. (a)-Shew that T(o, 1)7)–(0,0,0)" and T ((0,1,17)-(0,0,0)7. (b) Jstify whether T is en isomerphiem using result from (a) | (c) Given T(ē3) = (1,2, 3)". Find T (ei) and T (ē2) by using result from (a).
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