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t T: R³ R³ be the linear transformation defined by T(x, y, z) = (3x − 6y + 5z, x-2y+z, 2z) all (x, y, z) e R³. a) Find bases for the image and the kernel of T. >) Consider the ordered basis B = {(1, 2, 1), (1, 1, 2), (1, 2, -1)} of R³. Find m(T)B,B. c) Use (b) to determine if T is an isomorphism.

t T: R³ R³ be the linear transformation defined by T(x, y, z) = (3x − 6y + 5z, x-2y+z, 2z) all (x, y, z) e R³. a) Find bases for the image and the kernel of T. >) Consider the ordered basis B = {(1, 2, 1), (1, 1, 2), (1, 2, -1)} of R³. Find m(T)B,B. c) Use (b) to determine if T is an isomorphism.

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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Let T : R³ → R³ be the linear transformation defined by T(x, y, z) = (3x − 6y + 5z, x - 2y + z, 2z) for all (x, y, z) = R³. (a) Find bases for the image and the kernel of T. (b) Consider the ordered basis B = {(1, 2, 1), (1, 1, 2), (1, 2, −1)} of R³. Find m(T)‚· (c) Use (b) to determine if T is an isomorphism. (d) Let S be the standard ordered basis of R³. Determine the change-of-basis matrix from B to S. (e) Use (d) to find m(T)s,s.
Transcribed Image Text:Let T : R³ → R³ be the linear transformation defined by T(x, y, z) = (3x − 6y + 5z, x - 2y + z, 2z) for all (x, y, z) = R³. (a) Find bases for the image and the kernel of T. (b) Consider the ordered basis B = {(1, 2, 1), (1, 1, 2), (1, 2, −1)} of R³. Find m(T)‚· (c) Use (b) to determine if T is an isomorphism. (d) Let S be the standard ordered basis of R³. Determine the change-of-basis matrix from B to S. (e) Use (d) to find m(T)s,s.
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