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)T) = (0,0, 0)" whenever 2x – y + z = 0. (a)-Show that T(t,0, 1)7)–(0,0,O)T and T ((0,1,1)7"0,0,0)7- (b)Justify whether T-is an isomrorphism using result from (a). (c) Given T(e,)–(1,2,3)", Find Tle)and T(ë) by using result from(a) (d) Use your results of T(ēi),T(ē2),T(ē3) to determine the image of an arbitrary vector (x, y, z)T E R³ under T.
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)T) = (0,0, 0)" whenever 2x – y + z = 0. (a)-Show that T(t,0, 1)7)–(0,0,O)T and T ((0,1,1)7"0,0,0)7- (b)Justify whether T-is an isomrorphism using result from (a). (c) Given T(e,)–(1,2,3)", Find Tle)and T(ë) by using result from(a) (d) Use your results of T(ēi),T(ē2),T(ē3) to determine the image of an arbitrary vector (x, y, z)T E R³ under T.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 32E
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