1. (a) Use definition of linear transformation to determine whether the transformationT:R? → R³ defined as follows is linear.-yTr+ 2y2.r – 4y(b) Let T : R" → R" be a linear transformation.i. Is it always true that T(0) = 0 where the zero vectors live in the appropriatespaces? Justify.ii. Does S(0) = 0 mean that S is linear? If not, can you give a counter-example?(c) Consider the following matrix transformation T : R² → R² defined asa bTdi. If T rotates the 2D space 150° counter-clockwise, and then reflects the resultacross the new r-axis, find a, b, c, d.ii. What if T reflects the 2D space across the r-axis first and then rotates 150°counter-clockwise? Do you expect to get the same a, b, c, d? Why or why not? 2. (a) Solve the following system of linear equations by using Gauss-Jordan eliminationprocess.-11 + 3x2 – 2xz + 4.x44.x1 + 12x2 + 7.xz – 14.x4I1 - 3x2 + 4x3 - 87422.r1 – 6x2 + 13 – 2x4-3.Write the solution in column form by explicitly isolating the parameter(s) associatedwith free variable(s), if there is any.(b) Consider the following matrix1 4 -1 -32 71A =1 0-1 2 -3 -5i. Find an LU-factorization of A.ii. Use these L and U to solve the system Axb, where x =and7b-2(c) Find A-1. Use A-1 to solve the system Ax = b, where A, x and b are the same asgiven in 2(b).

Linear transformation:- A linear transformation or a linear map is a function T:ℝn→ℝm that satisfies…

Answered: 1. (a) Use definition of linear…

1. (a) Use definition of linear transformation to determine whether the transformation T: R - R defined as follows is linear. -y a+ 2y 2r – 4y

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

1. (a) Use definition of linear transformation to determine whether the transformation
T:R? → R³ defined as follows is linear.
-y
T
r+ 2y
2.r – 4y
(b) Let T : R" → R" be a linear transformation.
i. Is it always true that T(0) = 0 where the zero vectors live in the appropriate
spaces? Justify.
ii. Does S(0) = 0 mean that S is linear? If not, can you give a counter-example?
(c) Consider the following matrix transformation T : R² → R² defined as
a b
T
d
i. If T rotates the 2D space 150° counter-clockwise, and then reflects the result
across the new r-axis, find a, b, c, d.
ii. What if T reflects the 2D space across the r-axis first and then rotates 150°
counter-clockwise? Do you expect to get the same a, b, c, d? Why or why not?

2. (a) Solve the following system of linear equations by using Gauss-Jordan elimination
process.
-11 + 3x2 – 2xz + 4.x4
4.x1 + 12x2 + 7.xz – 14.x4
I1 - 3x2 + 4x3 - 874
2
2.r1 – 6x2 + 13 – 2x4
-3.
Write the solution in column form by explicitly isolating the parameter(s) associated
with free variable(s), if there is any.
(b) Consider the following matrix
1 4 -1 -3
2 7
1
A =
1 0
-1 2 -3 -5
i. Find an LU-factorization of A.
ii. Use these L and U to solve the system Ax
b, where x =
and
7
b
-2
(c) Find A-1. Use A-1 to solve the system Ax = b, where A, x and b are the same as
given in 2(b).

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