(a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 49 -1 X1 X1 + 2x2 - x3 X1 T| x2 4x1 + 9x2 - x3 i.e. T x2 X2 X3 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: 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.
(a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 49 -1 X1 X1 + 2x2 - x3 X1 T| x2 4x1 + 9x2 - x3 i.e. T x2 X2 X3 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: 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.
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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Question
please solve (b) part
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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