Having difficulty understanding what the Range of a linear transformation is.

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Discussion Overview

The discussion centers around the concept of the range of a linear transformation in linear algebra, specifically seeking clarification on its definition and relationship to the image of vectors under the transformation. Participants explore the distinctions between range, image, and codomain.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • One participant expresses a clear understanding of the kernel but finds the definition of the range nebulous, specifically questioning what it means to be "an image under T."
  • Another participant provides a response indicating that the range and image are synonymous, while also noting that the codomain is a separate concept.
  • A later reply acknowledges the initial explanation as helpful, suggesting it clarified the participant's confusion.

Areas of Agreement / Disagreement

There appears to be some agreement on the definitions of range and image being the same, but the distinction with codomain is noted as a point of clarification. The discussion does not resolve all uncertainties regarding these concepts.

Contextual Notes

The discussion does not address potential limitations in definitions or assumptions regarding surjectivity and its implications for the relationship between range and codomain.

phantomcow2
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One of the topics in my linear algebra course is kernel and range of a linear transformation. I have a firm understanding of what the kernel is: the set of vectors such that it maps all inputs to the zero vector. Range, however, remains nebulous to me. My textbook says that the range is "THe set of all vectors in W that are images under T of at least one vector in V."

I'm not sure what it means to be "an image under T." Could somebody explain this to me? I'd just like to have this concept clarified. Thanks.
 
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phantomcow2 said:
One of the topics in my linear algebra course is kernel and range of a linear transformation. I have a firm understanding of what the kernel is: the set of vectors such that it maps all inputs to the zero vector. Range, however, remains nebulous to me. My textbook says that the range is "THe set of all vectors in W that are images under T of at least one vector in V."

I'm not sure what it means to be "an image under T." Could somebody explain this to me? I'd just like to have this concept clarified. Thanks.

See the attachment.
 

Attachments

Wow, that is a wordy explanation and exactly what I needed. Thanks :).
 
The range and image are the same thing. But the codomain isn't. If a mapping is surjective, then they're all the same set, but in general, not.
 

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