Linear Algebra Topic |Let T:R' → R² be a linear transformation defined by(x, y, 2)T =(-x+3y+ z, y + 2z).(a) Do the following regarding the kemel of T, Ker T:(i) Describe the set that defines Ker Tin one statement (no working required).(ii) Find the set Si such that L(S) =Ker T (i.e. S¡ is the spanning set of Ker T}.(iii) Explain whether S, obtained above is linearly independent or linearly dependent.(iv) From your results above, conclude the basis and dimension of Ker T.(b) Do the following regarding the image of T, Im T :(i) Describe the set that defines Im Tin one statement (no working required).(ii) Find the set S, such that L(S2) = Im T and (i.e. S, is the spanning set of Im T}.(iii) Explain whether S2 obtained above is linearly independent or linearly dependent.(iv) From your results above, conclude the basis and dimension of Im T.(c) From your results in (a) and (b), evaluate whether Tis one-to-one or/and onto.

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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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Linear Algebra Topic

|Let T:R' → R² be a linear transformation defined by
(x, y, 2)T =(-x+3y+ z, y + 2z).
(a) Do the following regarding the kemel of T, Ker T:
(i) Describe the set that defines Ker Tin one statement (no working required).
(ii) Find the set Si such that L(S) =Ker T (i.e. S¡ is the spanning set of Ker T}.
(iii) Explain whether S, obtained above is linearly independent or linearly dependent.
(iv) From your results above, conclude the basis and dimension of Ker T.
(b) Do the following regarding the image of T, Im T :
(i) Describe the set that defines Im Tin one statement (no working required).
(ii) Find the set S, such that L(S2) = Im T and (i.e. S, is the spanning set of Im T}.
(iii) Explain whether S2 obtained above is linearly independent or linearly dependent.
(iv) From your results above, conclude the basis and dimension of Im T.
(c) From your results in (a) and (b), evaluate whether Tis one-to-one or/and onto.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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