Invertible 3x3 matrices a subspace of 3x3 matrices

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Homework Help Overview

The discussion revolves around whether the set of invertible 3x3 matrices qualifies as a subspace of the set of all 3x3 matrices. Participants are examining the properties that define a subspace in the context of linear algebra.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • One participant argues that the set of invertible matrices cannot be a subspace because the zero matrix is not included, while another participant confirms this reasoning. There is also a question regarding the definition of the 'neutral 0 element' and its relation to the additive identity.

Discussion Status

The discussion is exploring the definitions and properties related to subspaces, with participants affirming each other's reasoning. Questions about terminology and concepts are being raised, indicating an active engagement with the material.

Contextual Notes

Participants are considering the closure properties required for a subset to be classified as a subspace, specifically in relation to the additive identity and linear combinations.

wumple
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Homework Statement



Is the set of invertible 3x3 matrices a subspace of 3x3 matrices?

Homework Equations





The Attempt at a Solution


I think no - the 'neutral 0 element' is not in the subset since the 3x3 0 matrix is not in the subset. Am I right? The book says it's not a subspace because it's not closed under addition, but I'm not sure if my reason is also correct.
 
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Your reason is also correct.
 
thanks! also, quick question: does the 'neutral 0 element' mean the additive identity?
 
wumple said:
thanks! also, quick question: does the 'neutral 0 element' mean the additive identity?

Sure, subspace generally means closed under linear combinations. The zero matrix is the identity.
 

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