Linear Transformation S: Matrix A, Injective/Surjective

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Discussion Overview

The discussion revolves around the linear transformation S defined from R3 to R4, specifically focusing on finding the standard matrix A for S and determining whether S is injective or surjective. The context includes mathematical reasoning and homework-related inquiries.

Discussion Character

  • Homework-related, Mathematical reasoning

Main Points Raised

  • Post 1 presents the definition of the linear transformation S and asks for the standard matrix A and its injective/surjective properties.
  • Post 2 suggests that the problem appears to be a homework question and prompts the original poster to share their attempts.
  • Post 3 questions the notation "b1a1T" and similar terms, implying that these should be straightforward to interpret.
  • Post 4 reiterates the question about "b1a1T" and describes attempts to manipulate the formula for S(x), expressing uncertainty about obtaining the standard matrix and referencing a method involving the identity columns.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are multiple inquiries and uncertainties regarding the notation and the process for finding the standard matrix A. The discussion remains unresolved with respect to the injective or surjective nature of the transformation.

Contextual Notes

There are limitations in the clarity of the notation used and the steps taken to derive the standard matrix, as well as the application of the identity matrix in this context.

orange12
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Let the vectors a1,a2,a3 €R3 and b1,b2,b3 € R4 be given by

a1 a2 a3
1 -2 3
2 2 1
1 1 2

b1 b2 b3
1 1 -1
2 -3 2
1 4 3
3 -2 1

The linear transformation S : R3 --> R4 is defined by

S(x)= b1a1Tx+b2a2Tx+b3a3Tx x€R3

1. Find the standard matrix A for the linear transformation S og decide if the linear transformation S er injective or surjective.
 
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I smell homework. Tell us what you've tried.
 
What is "b1a1T" and the others? That should be easy to find.
 
HallsofIvy said:
What is "b1a1T" and the others? That should be easy to find.

I have tried to put the a1,a2 etc and b1,b2 etc into the formula for S(x).

First i put out x so it became

S(x)=(b1a1T+b2a2T+b3a3T)x, and then i get a Matrix, but i am not sure that it is the standard matrix. I read in my book that u have to see what it does to the Idendity colums, but can't figure out how to do that. I have tried out some things, but it would help a lot if you could show me which way is the right way to do it.
 

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