Task 1. Find the linear transformation : R" → R™ given by the following assignment -> a) [1,5] [3.-5], [2,-4] → [5,6] ->
Task 1. Find the linear transformation : R" → R™ given by the following assignment -> a) [1,5] [3.-5], [2,-4] → [5,6] ->
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.1: Introduction To Linear Transformations
Problem 39E: For the linear transformation from Exercise 33, find a T(1,1), b the preimage of (1,1), and c the...
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Please solve part a of each task. I promise I'll thumb up. Please solve part a from task 1,2,3,4,5.
I think you will do it soon. Please solve all.
and please do not provide solution in image format. thank you
![Task 1. Find the linear transformation : R→ RT given by the following assignment
[3,-5], [2, 4] → [5,6]
->
a) [1,5]
b) [3,1,1]
[4,3,3], [4, 1,4] -→ [5,9,-1], [5,1,3] ->
[4,3,3],[1,4] [5,9,-1]
c) [1,1]
->
Task 2. Investigate whether there exists a linear transformation p: R → R satisfying
-
a) [2, 4, 3] [1, 3], [1, 5, 4] → [0, 3], [9, 3, 1] → [7, 6]
-
[6,7,-3]
b) [1,0,3] [4, 5, 6], [4, 3, 1] → [3,8,-7],
c) [5, 4, 3] → [1, 0, 7], [3, 3, 3] → [2, 1, 5], [1, 2, 3] → [4, 2, 4].
Task 3. Determine the matrix of the linear transformation p: R" → R™ in the given bases
-
a)
([x, y, z]) = [x-z, 2x+2y-z], B = ([1, 0, 0], [0, -1,0], [0,0,- 1]), C =([-1, 0], [0, 1])
([x, y]) = [4x-3y, 9x-3x], B= ([1, 1], [1, -2]), C= ([2, 1], [-3, 1])
b)
c)
([x, y, z]) = [x+y, x+2y-3z], B = ([3, 1, 1], [5, -1,- 6], [4,-1,- 2]), C =([-1, 1], [1, 0])
Task 4 Write a linear transformation : R" → Rm if given a matrix of linear transformation
a) B= ([1,1],[1,2]), C = ([2,1], [-3,1]), M = [1 2 3 4],
b) B= ([1,0,0], [0, 1,0], [0,0,1]), C = ([-1, 0, 0], [0,-1,0], [0,0,-1]) M = [2 49 -
3484-71],
Task 5. For given linear transformations p: RR and y: RR find yo, using direct
substitution and confirm the results by multiplying the corresponding matrices
a) p ([x, y]) = [-x,x+y, -2x + y, 3x], y ([x, y, z, t]) = [x+z,y+z,z-t,t+x],
[x−z, 2x + 2y - z, z], y ([x,y,z]) = [x+z,x+y-z],
b) p ([x, y]) =
c) 9 ([x, y])
[x + 2,5x+z,z+y+x], y ([x,y,z]) = [3x-2y+z,x+2y-z],](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba0acd94-ebc3-41e6-80c8-3fde2b12e743%2Fd0698d51-d5dd-42dd-8cb7-6347acd6a9bf%2Fhuy1l12_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Task 1. Find the linear transformation : R→ RT given by the following assignment
[3,-5], [2, 4] → [5,6]
->
a) [1,5]
b) [3,1,1]
[4,3,3], [4, 1,4] -→ [5,9,-1], [5,1,3] ->
[4,3,3],[1,4] [5,9,-1]
c) [1,1]
->
Task 2. Investigate whether there exists a linear transformation p: R → R satisfying
-
a) [2, 4, 3] [1, 3], [1, 5, 4] → [0, 3], [9, 3, 1] → [7, 6]
-
[6,7,-3]
b) [1,0,3] [4, 5, 6], [4, 3, 1] → [3,8,-7],
c) [5, 4, 3] → [1, 0, 7], [3, 3, 3] → [2, 1, 5], [1, 2, 3] → [4, 2, 4].
Task 3. Determine the matrix of the linear transformation p: R" → R™ in the given bases
-
a)
([x, y, z]) = [x-z, 2x+2y-z], B = ([1, 0, 0], [0, -1,0], [0,0,- 1]), C =([-1, 0], [0, 1])
([x, y]) = [4x-3y, 9x-3x], B= ([1, 1], [1, -2]), C= ([2, 1], [-3, 1])
b)
c)
([x, y, z]) = [x+y, x+2y-3z], B = ([3, 1, 1], [5, -1,- 6], [4,-1,- 2]), C =([-1, 1], [1, 0])
Task 4 Write a linear transformation : R" → Rm if given a matrix of linear transformation
a) B= ([1,1],[1,2]), C = ([2,1], [-3,1]), M = [1 2 3 4],
b) B= ([1,0,0], [0, 1,0], [0,0,1]), C = ([-1, 0, 0], [0,-1,0], [0,0,-1]) M = [2 49 -
3484-71],
Task 5. For given linear transformations p: RR and y: RR find yo, using direct
substitution and confirm the results by multiplying the corresponding matrices
a) p ([x, y]) = [-x,x+y, -2x + y, 3x], y ([x, y, z, t]) = [x+z,y+z,z-t,t+x],
[x−z, 2x + 2y - z, z], y ([x,y,z]) = [x+z,x+y-z],
b) p ([x, y]) =
c) 9 ([x, y])
[x + 2,5x+z,z+y+x], y ([x,y,z]) = [3x-2y+z,x+2y-z],
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