Understanding Eigenspaces: Solving Exam Questions

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around the concept of eigenspaces in the context of preparing for exam questions. Participants explore the notation used to describe eigenspaces and clarify the relationship between eigenvalues and the corresponding eigenvectors.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether the notation "eigenspace 1, -1" is equivalent to "-1, 1," indicating a potential misunderstanding of the terms used.
  • Another participant suggests that the notation is confusing and proposes that it should refer to the subspace spanned by eigenvectors corresponding to the eigenvalues 1 and -1, emphasizing that the order of eigenvalues does not affect the subspace.
  • A further clarification is made that an eigenspace is a vector subspace, and the set {1, -1} does not represent a subspace, prompting a question about whether the participant meant the eigenspaces associated with the eigenvalues.
  • It is noted that if the notation refers to the vector <1, -1> in R2, then the spaces spanned by <1, -1> and <-1, 1> are indeed the same, as both vectors lie along the line y = -x but point in opposite directions.

Areas of Agreement / Disagreement

Participants express differing views on the clarity of the notation used to describe eigenspaces. While some agree on the equivalence of the spaces spanned by the vectors, there is no consensus on the initial notation and its implications.

Contextual Notes

The discussion highlights limitations in notation and the potential for confusion regarding the definitions of eigenspaces and eigenvectors. There is an unresolved aspect regarding the precise meaning of the notation used by the initial participant.

klingenberg
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I know this might be trivial, but when practicing for exam, I usually write the "inverse" values of the cheat sheet, and want to make sure I'm not making a mistake.

Is the eigenspace 1,-1 the same as -1,1?
 
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I think your difficulty is your "notation" which is so terse as to be confusing. 'The eigenspace 1, -1' makes no sense to me. If you mean "the subspace spanned by eigenvectors of linear operator A corresponding to eigenvalues 1 and -1" then it should be clear that the order in which you mention the eigenvalues is irrelevant.

(I started to write "the subspace of all eigenvectors of linear operator A corresponding to eigenvalues 1 and -1" but you understand that that set is NOT a subspace, right?)
 
To put it a bit differently, an eigenspace is a (vector ) subspace , and {1,-1} is not, at least not in any way I' familiar with. Did you mean the eigenspaces associated to each of these eigenvectors?
 
If your notation 1, -1 is intended to mean the vector <1, -1> in R2, then yes, the space spanned by the eigenvector <1, -1> is the same as that spanned by the vector <-1, 1>. Both vectors lie along the line y = -x but point in opposite directions.
 
Mark44 said:
If your notation 1, -1 is intended to mean the vector <1, -1> in R2, then yes, the space spanned by the eigenvector <1, -1> is the same as that spanned by the vector <-1, 1>. Both vectors lie along the line y = -x but point in opposite directions.

Perfect, thanks. Sorry about the confusing notation.
 

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