Determine whether the linear transformation T is one-to-one

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Homework Help Overview

The discussion revolves around determining whether specific linear transformations are one-to-one. The transformations in question involve polynomial functions, specifically from P2 to P3 and P2 to P2.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the concept of a linear transformation being one-to-one by discussing the kernel of the transformation and its implications. There are attempts to relate the problem to definitions and proofs previously learned, with some participants expressing difficulty due to a lack of practical examples.

Discussion Status

The discussion is ongoing, with participants sharing their struggles and seeking clarification on the concepts involved. Some guidance has been offered regarding the relationship between the kernel and one-to-one transformations, but there is no explicit consensus on how to proceed with the specific problems.

Contextual Notes

Participants note a lack of practical examples in their learning experience, which is affecting their confidence in tackling the problems presented. There is an emphasis on the need for practical applications of definitions and proofs in understanding the material.

hannahlu92
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Determine whether the linear transformation T is one-to-one

a) T:P2 --> P3, where T(a+a1x+a2x^2)=x(a+a1x+a2x^2)

b) T:P2 --> P2, where T(p(x))=p(x+1)

I'm having difficulty because my teacher never showed examples like this one.
Please help me on the procedure and solution.

 
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Hi hannahlu92! :smile:

Showing that an operator T is one-to-one is equivalent to showing that the kernel is 0. So, what you must show is that

T(a+bX+cX)=0

then a=b=c=0. Can you do that??
 
hannahlu92 said:
I'm having difficulty because my teacher never showed examples like this one.
Please help me on the procedure and solution.
So what if the problem looks different? Why can't you solve the problem the way you would normally do?

(Actually, it's probably easier to solve these problems than others you have faced, since there are easier ways than using your knowledge of linear algebra)
 
I don't know how to because I haven't done one before. My teacher writes definitions on the board and proofs, but no practical examples, so nothing is cementing in my brain.
 
hannahlu92 said:
I don't know how to because I haven't done one before. My teacher writes definitions on the board and proofs, but no practical examples, so nothing is cementing in my brain.

Certainly you don't need any practical examples to figure out when

T(a+bX+cX^2)=0

Just use the definition of T...
 
hannahlu92 said:
I don't know how to because I haven't done one before. My teacher writes definitions on the board and proofs, but no practical examples, so nothing is cementing in my brain.

In this case, proofs themselves are practical examples. That's the point of math courses.
 

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