Proving Non-Equivalence of Matrices Using Trace in Linear Algebra

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SUMMARY

This discussion focuses on proving the non-equivalence of matrices using the concept of trace in linear algebra. Participants explore the properties required for a subset U to be a subspace of V, specifically in the context of matrices in M3(ℝ). The proof confirms that U is non-empty, closed under vector addition, and closed under scalar multiplication, establishing U as a subspace. Additionally, it discusses the relationship between equivalence classes of matrices and their traces, concluding that each equivalence class can be uniquely labeled by its trace value.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly subspaces
  • Familiarity with matrix operations and properties
  • Knowledge of the trace function in linear algebra
  • Basic understanding of equivalence relations and classes
NEXT STEPS
  • Study the properties of subspaces in vector spaces
  • Learn about the trace function and its applications in linear algebra
  • Explore equivalence relations and their implications in mathematical structures
  • Investigate the dimension theorem in linear algebra and its applications
USEFUL FOR

Students and educators in linear algebra, mathematicians interested in matrix theory, and anyone looking to deepen their understanding of subspaces and equivalence relations in vector spaces.

  • #31
Yes, that seems to do it! It looks like you've got it!
 
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  • #32
micromass said:
Yes, that seems to do it! It looks like you've got it!

Or it may be better to write the second one as

[A]\neq<b> \iff A \not\sim B \iff A-B \not\in U \iff \text{trace}(A-B)\neq 0 \iff \text{trace}(A)\neq\text{trace}(B)</b>
 

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