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3. Let ē = (1,0,0)", 2 = (0,1,0)", s = (0,0, 1)" be the standard basis of R. Consider a linear map T : R* + R* satisfying T (r. y, 2)") = (0,0,0)" whenever 2x – y + z = 0. (a) Show that T (.0, – 1)") = (0,0,0)" and T (0, 1, 1)") = (0,0,0)". %3D %3D (b) Justify whether T is an isomorphism using result from (a). (c) Given T(es) = (1,2, 3)T. Find T () and T (2) by using result from (a). (d) Use vour results of T(e) Tiê) T(E) to determime the image of an arbitrary vector

3. Let ē = (1,0,0)", 2 = (0,1,0)", s = (0,0, 1)" be the standard basis of R. Consider a linear map T : R* + R* satisfying T (r. y, 2)") = (0,0,0)" whenever 2x – y + z = 0. (a) Show that T (.0, – 1)") = (0,0,0)" and T (0, 1, 1)") = (0,0,0)". %3D %3D (b) Justify whether T is an isomorphism using result from (a). (c) Given T(es) = (1,2, 3)T. Find T () and T (2) by using result from (a). (d) Use vour results of T(e) Tiê) T(E) to determime the image of an arbitrary vector

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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3. Let ē = (1,0,0)", 2 = (0, 1,0)", s = (0,0, 1)" be the standard basis of R*. Consider a linear map T : R* → R* satisfying T (r, y, 2)") = (0,0,0)" whenever 2x – y+ z = 0. (a) Show that T (. 0, – 1)") = (0,0,0)" and T ((0, 1, 1)") = (0,0,0)". %3D %3D (b) Justify whether T is an isomorphism using result from (a). (c) Given T(ēs) = (1,2, 3)T. Find T (č) and T () by using result from (a). (d) Use your results of T(ei), T(ē2), T(ēs) to determine the image of an arbitrary vector (r, y, 2)T E R under T.
Transcribed Image Text:3. Let ē = (1,0,0)", 2 = (0, 1,0)", s = (0,0, 1)" be the standard basis of R*. Consider a linear map T : R* → R* satisfying T (r, y, 2)") = (0,0,0)" whenever 2x – y+ z = 0. (a) Show that T (. 0, – 1)") = (0,0,0)" and T ((0, 1, 1)") = (0,0,0)". %3D %3D (b) Justify whether T is an isomorphism using result from (a). (c) Given T(ēs) = (1,2, 3)T. Find T (č) and T () by using result from (a). (d) Use your results of T(ei), T(ē2), T(ēs) to determine the image of an arbitrary vector (r, y, 2)T E R under T.
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