Is T(inv)(0) a Subspace of V and T(V) a Subspace of W?

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Discussion Overview

The discussion revolves around whether the set T(inv)(0) is a subspace of vector space V and whether T(V) is a subspace of vector space W, given a linear map T: V -> W. The scope includes theoretical aspects of linear algebra and properties of vector spaces.

Discussion Character

  • Homework-related
  • Technical explanation

Main Points Raised

  • One participant requests assistance in verifying if T(inv)(0) is a subspace of V and T(V) is a subspace of W.
  • Another participant suggests checking the axioms for a subspace to determine if the conditions are satisfied.
  • A third participant inquires if the three axioms for subspaces are the focus of the verification.
  • A later reply emphasizes the importance of posting any progress made in the verification process rather than seeking complete solutions.
  • There is a suggestion that something qualifies as a subspace if and only if it meets the necessary rules for subspaces.

    • Areas of Agreement / Disagreement

    Participants generally agree on the need to verify the axioms for subspaces, but there is no consensus on the specific progress made or the level of detail required in the responses.

    Contextual Notes

    Limitations include potential missing assumptions regarding the definitions of the vector spaces and the linear map, as well as the specific axioms being referenced in the discussion.

    Who May Find This Useful

    Students or individuals studying linear algebra, particularly those interested in the properties of vector spaces and linear mappings.

mang733
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Could someone help me here please?

Let V and W be two vector spaces and T: V -> W be a linear map. Define

T(inv) (0) = { u element of V l T(u) = 0 }

where 0 is the zero element in W. Also define,

T(V) = { T(u) l u element of V},

the image of V under T. Show that T(inv) (0) is a subspace of V and T(V) is a subspace of W.

Your help is much appreciated!
 
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Just chekc the axioms for a subspave are satisfied, and post your working.
 
you mean the three axioms?
 
since I want to make sure I have the correct answer, if someone has the time I appreciate a detailed response.
 
Doing someone else's homework isn't very interesting to most people. So post what you've managed to verify. Something is a subspace if and only iof it satisfies the rules for being a subspace; how far have you been able to verify that the rules hold?
 

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