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(a) Consider the real linear map T : R' - R', expressed as follows: X1 + 2x2 - x3 4x1 + 9x2 - x3 1 2 -1 49 -1 X1 X1 T i.e. TX2 %3D %3D X2 X2 2x1 +7x2 +7x3, X3 27 7 (i) Find a basis for the kernel of T. (ii) Find a basis for the image of T. (iii) Does the image of T include every vector in R'? Justity your answer. (You are not required to verify that your answers to (a)(i), (a)(i) are bases.)

(a) Consider the real linear map T : R' - R', expressed as follows: X1 + 2x2 - x3 4x1 + 9x2 - x3 1 2 -1 49 -1 X1 X1 T i.e. TX2 %3D %3D X2 X2 2x1 +7x2 +7x3, X3 27 7 (i) Find a basis for the kernel of T. (ii) Find a basis for the image of T. (iii) Does the image of T include every vector in R'? Justity your answer. (You are not required to verify that your answers to (a)(i), (a)(i) are bases.)

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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(a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 4 9 -1 X1 + 2x2 - x3 X1 X1 X1 T| x2 4x1 + 9x2 - x3 i.e. T x2 X2 2x1 + 7x2 + 7x3) X3 2 7 7 X3 (i) Find a basis for the kernel of T. (ii) Find a basis for the image of T. (iii) Does the image of T include every vector in R'? Justify your answer. (You are not required to verify that your answers to (a) (i), (a) (i) are bases.) (b) Consider the following subset of R: X1 S = x2 ER: x2 + 4x3 = 2 Does there exist a real linear map, from R to R' say, such that the set S is the image of the real linear map? Justify your answer.
Transcribed Image Text:(a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 4 9 -1 X1 + 2x2 - x3 X1 X1 X1 T| x2 4x1 + 9x2 - x3 i.e. T x2 X2 2x1 + 7x2 + 7x3) X3 2 7 7 X3 (i) Find a basis for the kernel of T. (ii) Find a basis for the image of T. (iii) Does the image of T include every vector in R'? Justify your answer. (You are not required to verify that your answers to (a) (i), (a) (i) are bases.) (b) Consider the following subset of R: X1 S = x2 ER: x2 + 4x3 = 2 Does there exist a real linear map, from R to R' say, such that the set S is the image of the real linear map? Justify your answer.
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